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

<span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">L</font> being  <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/> for <font color="Olive" title="b2">d</font> being  <a href="lattice5.html#NM3" title="LATTICE5:NM.3">BiFunction</a> of <font color="Maroon" title="c1">A</font>,<font color="Olive" title="b1">L</font>  st <font color="Olive" title="b2">d</font> is  <a href="lattice5.html#V1" title="LATTICE5:attr.1">symmetric</a>  &amp; <font color="Olive" title="b2">d</font> is  <a href="lattice5.html#V3" title="LATTICE5:attr.3">u.t.i.</a>  holds <br/>for <font color="Olive" title="b3">q</font> being   <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="Olive" title="b1">L</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">L</font></span><a href="zfmisc_1.html#K4" title="ZFMISC_1:func.4">:]</a></span>  st <font color="Olive" title="b2">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">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="Olive" title="b3">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="Olive" title="b3">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="Olive" title="b3">q</font> <a href="mcart_1.html#K11" title="MCART_1:func.11">`4</a> </span>)</span> holds <br/> <a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Olive" title="b2">d</font>,<font color="Olive" title="b3">q</font> is  <a href="lattice5.html#V3" title="LATTICE5:attr.3">u.t.i.</a> </span><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">d</font> being  <a href="lattice5.html#NM3" title="LATTICE5:NM.3">BiFunction</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">L</font>  st <font color="Olive" title="b1">d</font> is  <a href="lattice5.html#V1" title="LATTICE5:attr.1">symmetric</a>  &amp; <font color="Olive" title="b1">d</font> is  <a href="lattice5.html#V3" title="LATTICE5:attr.3">u.t.i.</a>  holds <br/>for <font color="Olive" title="b2">q</font> being   <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>  st <font color="Olive" title="b1">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">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="Olive" title="b2">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="Olive" title="b2">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="Olive" title="b2">q</font> <a href="mcart_1.html#K11" title="MCART_1:func.11">`4</a> </span>)</span> holds <br/> <a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Olive" title="b1">d</font>,<font color="Olive" title="b2">q</font> is  <a href="lattice5.html#V3" title="LATTICE5:attr.3">u.t.i.</a> </span><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">d</font> is  <a href="lattice5.html#V1" title="LATTICE5:attr.1">symmetric</a>  &amp; <font color="Maroon" title="c3">d</font> is  <a href="lattice5.html#V3" title="LATTICE5:attr.3">u.t.i.</a>  implies for <font color="Olive" title="b1">q</font> being   <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>  st <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="Olive" title="b1">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="Olive" title="b1">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="Olive" title="b1">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="Olive" title="b1">q</font> <a href="mcart_1.html#K11" title="MCART_1:func.11">`4</a> </span>)</span> holds <br/> <a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Olive" title="b1">q</font> is  <a href="lattice5.html#V3" title="LATTICE5:attr.3">u.t.i.</a>  )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:42"/><span class="lab"><font color="Green" title="E27">A1</font></span>: 
<font color="Maroon" title="c3">d</font> is  <a href="lattice5.html#V1" title="LATTICE5:attr.1">symmetric</a> 
 <span class="kw">and </span><br/><a NAME="E2:42"/><span class="lab"><font color="Green" title="E28">A2</font></span>: 
<font color="Maroon" title="c3">d</font> is  <a href="lattice5.html#V3" title="LATTICE5:attr.3">u.t.i.</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">q</font> being   <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>  st <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="Olive" title="b1">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="Olive" title="b1">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="Olive" title="b1">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="Olive" title="b1">q</font> <a href="mcart_1.html#K11" title="MCART_1:func.11">`4</a> </span>)</span> holds <br/> <a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Olive" title="b1">q</font> is  <a href="lattice5.html#V3" title="LATTICE5:attr.3">u.t.i.</a> </span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c4">B</font> = <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1: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 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 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"><span class="p4"><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#K1" title="TARSKI:func.1">}</a></span></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</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/>
<span class="kw">let </span><font color="Maroon" title="c5">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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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="c5">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="c5">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="c5">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="c5">q</font> <a href="mcart_1.html#K11" title="MCART_1:func.11">`4</a> </span>)</span> implies  <a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font> is  <a href="lattice5.html#V3" title="LATTICE5:attr.3">u.t.i.</a>  )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c6">x</font> = <font color="Maroon" title="c5">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">y</font> = <font color="Maroon" title="c5">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">f</font> =  <a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">a</font> = <font color="Maroon" title="c5">q</font> <a href="mcart_1.html#K10" title="MCART_1:func.10">`3</a> , <font color="Maroon" title="c10">b</font> = <font color="Maroon" title="c5">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="E5:42"/><span class="lab"><font color="Green" title="E29">A3</font></span>: 
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="lattice5.html#K6" title="LATTICE5:func.6">new_set</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="c4">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">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="42_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">p</font>, <font color="Maroon" title="c12">q</font>, <font color="Maroon" title="c13">u</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice5.html#K6" title="LATTICE5:func.6">new_set</a> <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">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="c12">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> implies <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:42_1"/><span class="lab"><font color="Green" title="E30">A4</font></span>: 
( <font color="Maroon" title="c11">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="c12">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:42_1_1"/>
( ( <font color="Maroon" title="c11">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="c13">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="c12">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="c11">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="c13">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="c12">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> ) or ( <font color="Maroon" title="c11">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="c13">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="c12">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"><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#K1" title="TARSKI:func.1">}</a></span> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_1"><span class="lab"><font color="Green" title="E30">A4</font></span></a>, <a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:42_1_1_1"/><span class="lab"><font color="Green" title="E31">A5</font></span>: 
( <font color="Maroon" title="c11">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="c13">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="c12">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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">p9</font> = <font color="Maroon" title="c11">p</font>, <font color="Maroon" title="c15">u9</font> = <font color="Maroon" title="c13">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="E3:42_1_1_1"/>
<font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p9</font>,<font color="Maroon" title="c15">u9</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="c14">p9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c5">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<font color="Maroon" title="c15">u9</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="ref" href="lattice5.html#D8" target="_self" title="LATTICE5:def.8">Def8</a></span>;<br/><a NAME="E4:42_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A6</font></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="c14">p9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c15">u9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c14">p9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c15">u9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font> &amp; <font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p9</font>,<font color="Maroon" title="c15">u9</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="c14">p9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c15">u9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="ref" href="lattice5.html#D6" target="_self" title="LATTICE5:def.6">Def6</a>, <a class="ref" href="yellow_0.html#T22" title="YELLOW_0:th.22">YELLOW_0:22</a></span>;<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="c14">p9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c15">u9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c9">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="c14">p9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c15">u9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></span>
<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="c14">p9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c15">u9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_5.html#T1" title="YELLOW_5:th.1">YELLOW_5:1</a></span>
<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="c14">p9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c15">u9</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></span>
<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="c14">p9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c9">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="c15">u9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></span>
;<br/><a NAME="E6:42_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A7</font></span>: 
<font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p9</font>,<font color="Maroon" title="c15">u9</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="c14">p9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c9">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="c15">u9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:42_1_1_1"><span class="lab"><font color="Green" title="E32">A6</font></span></a>, <a class="ref" href="orders_2.html#T26" title="ORDERS_2:th.26">ORDERS_2:26</a></span>;<br/><a NAME="E7:42_1_1_1"/>
( <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="c14">p9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font> &amp; <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">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="c15">u9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_1_1_1"><span class="lab"><font color="Green" title="E31">A5</font></span></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>;<br/><span class="kw">hence </span><a NAME="E8:42_1_1_1"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:42_1_1_1"><span class="lab"><font color="Green" title="E33">A7</font></span></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:42_1_1_2"/><span class="lab"><font color="Green" title="E31">A8</font></span>: 
( <font color="Maroon" title="c11">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="c13">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="c12">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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">p9</font> = <font color="Maroon" title="c11">p</font>, <font color="Maroon" title="c15">u9</font> = <font color="Maroon" title="c13">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="E3:42_1_1_2"/>
<font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p9</font>,<font color="Maroon" title="c15">u9</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="c14">p9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c5">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<font color="Maroon" title="c15">u9</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="ref" href="lattice5.html#D8" target="_self" title="LATTICE5:def.8">Def8</a></span>;<br/><a NAME="E4:42_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A9</font></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="c14">p9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c15">u9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c14">p9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c15">u9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> &amp; <font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p9</font>,<font color="Maroon" title="c15">u9</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="c14">p9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c15">u9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="ref" href="lattice5.html#D6" target="_self" title="LATTICE5:def.6">Def6</a>, <a class="ref" href="yellow_0.html#T22" title="YELLOW_0:th.22">YELLOW_0:22</a></span>;<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="c14">p9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c15">u9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</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="c14">p9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c15">u9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></span>
<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="c14">p9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c15">u9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_5.html#T1" title="YELLOW_5:th.1">YELLOW_5:1</a></span>
<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="c14">p9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c15">u9</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></span>
<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="c14">p9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></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="c15">u9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></span>
<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="c14">p9</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">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"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c15">u9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></span>
<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="c14">p9</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">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"><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="c15">u9</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></span>
;<br/><a NAME="E6:42_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A10</font></span>: 
<font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p9</font>,<font color="Maroon" title="c15">u9</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"><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="c14">p9</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">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"><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="c15">u9</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:42_1_1_2"><span class="lab"><font color="Green" title="E32">A9</font></span></a>, <a class="ref" href="orders_2.html#T26" title="ORDERS_2:th.26">ORDERS_2:26</a></span>;<br/><a NAME="E7:42_1_1_2"/>
( <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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> <font color="Maroon" title="c14">p9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font> &amp; <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">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> <font color="Maroon" title="c15">u9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_1_1_2"><span class="lab"><font color="Green" title="E31">A8</font></span></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>;<br/><span class="kw">hence </span><a NAME="E8:42_1_1_2"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:42_1_1_2"><span class="lab"><font color="Green" title="E33">A10</font></span></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:42_1_1_3"/><span class="lab"><font color="Green" title="E31">A11</font></span>: 
( <font color="Maroon" title="c11">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="c13">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="c12">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"><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#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">p9</font> = <font color="Maroon" title="c11">p</font>, <font color="Maroon" title="c15">u9</font> = <font color="Maroon" title="c13">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="E3:42_1_1_3"/>
<font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p9</font>,<font color="Maroon" title="c15">u9</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="c14">p9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c5">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span>,<font color="Maroon" title="c15">u9</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="ref" href="lattice5.html#D8" target="_self" title="LATTICE5:def.8">Def8</a></span>;<br/><a NAME="E4:42_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A12</font></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="c14">p9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c15">u9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c14">p9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c15">u9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c10">b</font> &amp; <font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p9</font>,<font color="Maroon" title="c15">u9</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="c14">p9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c15">u9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="ref" href="lattice5.html#D6" target="_self" title="LATTICE5:def.6">Def6</a>, <a class="ref" href="yellow_0.html#T22" title="YELLOW_0:th.22">YELLOW_0:22</a></span>;<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="c14">p9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c15">u9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c10">b</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="c14">p9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c15">u9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c10">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></span>
<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="c14">p9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c15">u9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c10">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_5.html#T1" title="YELLOW_5:th.1">YELLOW_5:1</a></span>
<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="c14">p9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c15">u9</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c5">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="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></span>
<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="c14">p9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c10">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"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c15">u9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c10">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></span>
;<br/><a NAME="E6:42_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A13</font></span>: 
<font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p9</font>,<font color="Maroon" title="c15">u9</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="c14">p9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c10">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"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c15">u9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c10">b</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:42_1_1_3"><span class="lab"><font color="Green" title="E32">A12</font></span></a>, <a class="ref" href="orders_2.html#T26" title="ORDERS_2:th.26">ORDERS_2:26</a></span>;<br/><a NAME="E7:42_1_1_3"/>
( <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="c14">p9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c10">b</font> &amp; <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">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="c15">u9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c10">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_1_1_3"><span class="lab"><font color="Green" title="E31">A11</font></span></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>;<br/><span class="kw">hence </span><a NAME="E8:42_1_1_3"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:42_1_1_3"><span class="lab"><font color="Green" title="E33">A13</font></span></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<span class="kw">assume </span><a NAME="E6:42"/><span class="lab"><font color="Green" title="E30">A14</font></span>: 
<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="c5">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="c5">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="c5">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="c5">q</font> <a href="mcart_1.html#K11" title="MCART_1:func.11">`4</a> </span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font> is  <a href="lattice5.html#V3" title="LATTICE5:attr.3">u.t.i.</a> </span><br/>

<a NAME="E7:42"/><span class="lab"><font color="Green" title="E31">A15</font></span>: 
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="lattice5.html#K6" title="LATTICE5:func.6">new_set</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="c4">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="c4">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">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="42_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">p</font>, <font color="Maroon" title="c12">q</font>, <font color="Maroon" title="c13">u</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice5.html#K6" title="LATTICE5:func.6">new_set</a> <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> implies <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:42_2"/><span class="lab"><font color="Green" title="E32">A16</font></span>: 
( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> )
 <span class="kw">and </span><br/><a NAME="E2:42_2"/><span class="lab"><font color="Green" title="E33">A17</font></span>: 
<font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c14">u9</font> = <font color="Maroon" title="c13">u</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42_2"><span class="lab"><font color="Green" title="E33">A17</font></span></a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:42_2_1"/>
( ( <font color="Maroon" title="c13">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="c12">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="c11">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="c13">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="c12">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="c11">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="c13">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="c12">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="c11">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"><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#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c13">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="c12">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="c11">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="c13">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="c12">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="c11">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="c13">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="c12">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="c11">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"><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#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c13">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="c12">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"><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#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c11">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="c13">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="c12">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"><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#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c11">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="c13">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="c12">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"><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#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c11">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"><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#K1" title="TARSKI:func.1">}</a></span> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_2"><span class="lab"><font color="Green" title="E32">A16</font></span></a>, <a class="txt" href="#E2:42_2"><span class="lab"><font color="Green" title="E33">A17</font></span></a>, <a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:42_2_1_1"/><span class="lab"><font color="Green" title="E34">A18</font></span>: 
( <font color="Maroon" title="c13">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="c12">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="c11">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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><span class="kw">then </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_2_1_1"><span class="lab"><font color="Green" title="E34">A18</font></span></a>, <a class="ref" href="waybel_1.html#T4" title="WAYBEL_1:th.4">WAYBEL_1:4</a></span>
;<br/><span class="kw">hence </span><a NAME="E3:42_2_1_1"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:42_2_1_2"/><span class="lab"><font color="Green" title="E34">A19</font></span>: 
( <font color="Maroon" title="c13">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="c12">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="c11">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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><span class="kw">then </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span> = 
<font color="Maroon" title="c10">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>
<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="c14">u9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_2_1_2"><span class="lab"><font color="Green" title="E34">A19</font></span></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>
;<br/><span class="kw">hence </span><a NAME="E3:42_2_1_2"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_2_1_2"><span class="lab"><font color="Green" title="E34">A19</font></span></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_2_1_3"><span class="kw">suppose </span></a><a NAME="E1:42_2_1_3"/><span class="lab"><font color="Green" title="E34">A20</font></span>: 
( <font color="Maroon" title="c13">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="c12">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="c11">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"><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#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><font color="Maroon" title="c10">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></span>
<br/>.= 
<font color="Maroon" title="c10">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_5.html#T1" title="YELLOW_5:th.1">YELLOW_5:1</a></span>
<br/>.= 
<font color="Maroon" title="c10">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_5.html#T1" title="YELLOW_5:th.1">YELLOW_5:1</a></span>
<br/>.= 
<font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></span>
;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A21</font></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="c14">u9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</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="c14">u9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">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"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">u9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">u9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_2_1_3"><span class="lab"><font color="Green" title="E34">A20</font></span></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>
;<br/><a NAME="E4:42_2_1_3"/>
<font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">u9</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">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="c14">u9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c5">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="c5">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="ref" href="lattice5.html#D8" target="_self" title="LATTICE5:def.8">Def8</a></span>;<br/><a NAME="E5:42_2_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A22</font></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="c14">u9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c10">b</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="c14">u9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c5">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="c5">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="c10">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_1.html#T3" title="WAYBEL_1:th.3">WAYBEL_1:3</a></span>;<br/><a NAME="E6:42_2_1_3"/>
<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="c14">u9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c10">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> <font color="Maroon" title="c14">u9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c10">b</font>
 ;<br/><a NAME="E7:42_2_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A23</font></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="c14">u9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c5">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="c5">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="c10">b</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> <font color="Maroon" title="c14">u9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c10">b</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> <span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c5">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> &amp; <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="c14">u9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c10">b</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> <span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c5">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="c14">u9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:42"><span class="lab"><font color="Green" title="E30">A14</font></span></a>, <a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a>, <a class="ref" href="yellow_3.html#T3" title="YELLOW_3:th.3">YELLOW_3:3</a></span>;<br/><a NAME="E8:42_2_1_3"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="c14">u9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c10">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_2_1_3"><span class="lab"><font color="Green" title="E34">A20</font></span></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>;<br/><a NAME="E9:42_2_1_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="c14">u9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:42_2_1_3"><span class="lab"><font color="Green" title="E36">A22</font></span></a>, <a class="txt" href="#E7:42_2_1_3"><span class="lab"><font color="Green" title="E37">A23</font></span></a>, <a class="ref" href="orders_2.html#T26" title="ORDERS_2:th.26">ORDERS_2:26</a></span>;<br/><span class="kw">hence </span><a NAME="E10:42_2_1_3"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_2_1_3"><span class="lab"><font color="Green" title="E34">A20</font></span></a>, <a class="txt" href="#E3:42_2_1_3"><span class="lab"><font color="Green" title="E35">A21</font></span></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_2_1_4"><span class="kw">suppose </span></a><a NAME="E1:42_2_1_4"/><span class="lab"><font color="Green" title="E34">A24</font></span>: 
( <font color="Maroon" title="c13">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="c12">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="c11">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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><span class="kw">then </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span> = 
<font color="Maroon" title="c10">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>
<br/>.= 
<font color="Maroon" title="c10">b</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> <font color="Maroon" title="c14">u9</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_2_1_4"><span class="lab"><font color="Green" title="E34">A24</font></span></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>
<br/>.= 
<font color="Maroon" title="c10">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_2_1_4"><span class="lab"><font color="Green" title="E34">A24</font></span></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></span>
<br/>.= 
<font color="Maroon" title="c10">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_5.html#T1" title="YELLOW_5:th.1">YELLOW_5:1</a></span>
;<br/><span class="kw">hence </span><a NAME="E3:42_2_1_4"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T22" title="YELLOW_0:th.22">YELLOW_0:22</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_2_1_5"><span class="kw">suppose </span></a><a NAME="E1:42_2_1_5"/><span class="lab"><font color="Green" title="E34">A25</font></span>: 
( <font color="Maroon" title="c13">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="c12">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="c11">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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><span class="kw">then </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_2_1_5"><span class="lab"><font color="Green" title="E34">A25</font></span></a>, <a class="ref" href="waybel_1.html#T4" title="WAYBEL_1:th.4">WAYBEL_1:4</a></span>
;<br/><span class="kw">hence </span><a NAME="E3:42_2_1_5"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_2_1_6"><span class="kw">suppose </span></a><a NAME="E1:42_2_1_6"/><span class="lab"><font color="Green" title="E34">A26</font></span>: 
( <font color="Maroon" title="c13">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="c12">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="c11">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"><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#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><font color="Maroon" title="c10">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></span>
<br/>.= 
<font color="Maroon" title="c10">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_5.html#T1" title="YELLOW_5:th.1">YELLOW_5:1</a></span>
<br/>.= 
<font color="Maroon" title="c10">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_5.html#T1" title="YELLOW_5:th.1">YELLOW_5:1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></span>
;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A27</font></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="c14">u9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</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="c14">u9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></span>
<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="c14">u9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></span>
<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="c14">u9</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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"><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="c14">u9</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_2_1_6"><span class="lab"><font color="Green" title="E34">A26</font></span></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>
;<br/><a NAME="E4:42_2_1_6"/>
<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="c14">u9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c10">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> <font color="Maroon" title="c14">u9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c10">b</font>
 ;<br/><a NAME="E5:42_2_1_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A28</font></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="c14">u9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c5">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="c5">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="c10">b</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> <font color="Maroon" title="c14">u9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c10">b</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> <span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c5">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> &amp; <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="c14">u9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c10">b</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> <span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c5">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="c14">u9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:42"><span class="lab"><font color="Green" title="E30">A14</font></span></a>, <a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a>, <a class="ref" href="yellow_3.html#T3" title="YELLOW_3:th.3">YELLOW_3:3</a></span>;<br/><a NAME="E6:42_2_1_6"/>
<font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">u9</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">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="c14">u9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c5">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="c5">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="ref" href="lattice5.html#D8" target="_self" title="LATTICE5:def.8">Def8</a></span>;<br/><a NAME="E7:42_2_1_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A29</font></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="c14">u9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c10">b</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="c14">u9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c5">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="c5">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="c10">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_1.html#T3" title="WAYBEL_1:th.3">WAYBEL_1:3</a></span>;<br/><a NAME="E8:42_2_1_6"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="c14">u9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c10">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_2_1_6"><span class="lab"><font color="Green" title="E34">A26</font></span></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>;<br/><a NAME="E9:42_2_1_6"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="c14">u9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:42_2_1_6"><span class="lab"><font color="Green" title="E37">A29</font></span></a>, <a class="txt" href="#E5:42_2_1_6"><span class="lab"><font color="Green" title="E36">A28</font></span></a>, <a class="ref" href="orders_2.html#T26" title="ORDERS_2:th.26">ORDERS_2:26</a></span>;<br/><span class="kw">hence </span><a NAME="E10:42_2_1_6"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_2_1_6"><span class="lab"><font color="Green" title="E34">A26</font></span></a>, <a class="txt" href="#E3:42_2_1_6"><span class="lab"><font color="Green" title="E35">A27</font></span></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_2_1_7"><span class="kw">suppose </span></a><a NAME="E1:42_2_1_7"/><span class="lab"><font color="Green" title="E34">A30</font></span>: 
( <font color="Maroon" title="c13">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="c12">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"><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#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c11">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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><span class="lab"><font color="Green" title="E35">A31</font></span>: <font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></span>
<br/>.= 
<font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_5.html#T1" title="YELLOW_5:th.1">YELLOW_5:1</a></span>
<br/>.= 
<font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_5.html#T1" title="YELLOW_5:th.1">YELLOW_5:1</a></span>
<br/>.= 
<font color="Maroon" title="c10">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></span>
;<br/><span class="lab"><font color="Green" title="E36">A32</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c9">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> <font color="Maroon" title="c14">u9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</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="c14">u9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></span>
<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="c14">u9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42_2_1_7"><span class="lab"><font color="Green" title="E35">A31</font></span></a>, <a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">u9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c10">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_2_1_7"><span class="lab"><font color="Green" title="E34">A30</font></span></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>
;<br/><a NAME="E4:42_2_1_7"/>
<font color="Maroon" title="c9">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> <font color="Maroon" title="c14">u9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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> <font color="Maroon" title="c9">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> <font color="Maroon" title="c14">u9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span>
 ;<br/><a NAME="E5:42_2_1_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A33</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">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> <font color="Maroon" title="c14">u9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c5">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"><font color="Maroon" title="c9">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> <font color="Maroon" title="c14">u9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:42"><span class="lab"><font color="Green" title="E30">A14</font></span></a>, <a class="ref" href="yellow_3.html#T3" title="YELLOW_3:th.3">YELLOW_3:3</a></span>;<br/><a NAME="E6:42_2_1_7"/>
<font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">u9</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">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="c14">u9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c5">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="ref" href="lattice5.html#D8" target="_self" title="LATTICE5:def.8">Def8</a></span>;<br/><a NAME="E7:42_2_1_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A34</font></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="c14">u9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c9">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="c14">u9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c5">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_1.html#T3" title="WAYBEL_1:th.3">WAYBEL_1:3</a></span>;<br/><span class="lab"><font color="Green" title="E39">A35</font></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="c14">u9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c5">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c9">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> <span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c14">u9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">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> <font color="Maroon" title="c14">u9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c5">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="ref" href="lattice5.html#D6" target="_self" title="LATTICE5:def.6">Def6</a></span>
;<br/><a NAME="E9:42_2_1_7"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="c14">u9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_2_1_7"><span class="lab"><font color="Green" title="E34">A30</font></span></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>;<br/><a NAME="E10:42_2_1_7"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="c9">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> <font color="Maroon" title="c14">u9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:42_2_1_7"><span class="lab"><font color="Green" title="E38">A34</font></span></a>, <a class="txt" href="#E8:42_2_1_7"><span class="lab"><font color="Green" title="E39">A35</font></span></a>, <a class="txt" href="#E5:42_2_1_7"><span class="lab"><font color="Green" title="E37">A33</font></span></a>, <a class="ref" href="orders_2.html#T26" title="ORDERS_2:th.26">ORDERS_2:26</a></span>;<br/><span class="kw">hence </span><a NAME="E11:42_2_1_7"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_2_1_7"><span class="lab"><font color="Green" title="E34">A30</font></span></a>, <a class="txt" href="#E3:42_2_1_7"><span class="lab"><font color="Green" title="E36">A32</font></span></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_2_1_8"><span class="kw">suppose </span></a><a NAME="E1:42_2_1_8"/><span class="lab"><font color="Green" title="E34">A36</font></span>: 
( <font color="Maroon" title="c13">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="c12">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"><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#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c11">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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A37</font></span>: <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> = 
<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="c14">u9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>
<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="c14">u9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></span>
;<br/><a NAME="E3:42_2_1_8"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</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="c14">u9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</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="c14">u9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span>
 ;<br/><a NAME="E4:42_2_1_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A38</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</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="c14">u9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c5">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="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</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="c14">u9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:42"><span class="lab"><font color="Green" title="E30">A14</font></span></a>, <a class="ref" href="yellow_3.html#T3" title="YELLOW_3:th.3">YELLOW_3:3</a></span>;<br/><a NAME="E5:42_2_1_8"/>
<font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">u9</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">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="c14">u9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c5">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="ref" href="lattice5.html#D8" target="_self" title="LATTICE5:def.8">Def8</a></span>;<br/><a NAME="E6:42_2_1_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A39</font></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="c14">u9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</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> <font color="Maroon" title="c14">u9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c5">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_1.html#T3" title="WAYBEL_1:th.3">WAYBEL_1:3</a></span>;<br/><span class="lab"><font color="Green" title="E38">A40</font></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="c14">u9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c5">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</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="c14">u9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</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="c14">u9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c5">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="ref" href="lattice5.html#D6" target="_self" title="LATTICE5:def.6">Def6</a></span>
;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span> = 
<font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_2_1_8"><span class="lab"><font color="Green" title="E34">A36</font></span></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>
<br/>.= 
<font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">b</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> <font color="Maroon" title="c14">u9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_2_1_8"><span class="lab"><font color="Green" title="E34">A36</font></span></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</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="c14">u9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></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="c14">u9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_5.html#T1" title="YELLOW_5:th.1">YELLOW_5:1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">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"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">u9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></span>
;<br/><span class="kw">hence </span><a NAME="E9:42_2_1_8"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42_2_1_8"><span class="lab"><font color="Green" title="E35">A37</font></span></a>, <a class="txt" href="#E6:42_2_1_8"><span class="lab"><font color="Green" title="E37">A39</font></span></a>, <a class="txt" href="#E7:42_2_1_8"><span class="lab"><font color="Green" title="E38">A40</font></span></a>, <a class="txt" href="#E4:42_2_1_8"><span class="lab"><font color="Green" title="E36">A38</font></span></a>, <a class="ref" href="orders_2.html#T26" title="ORDERS_2:th.26">ORDERS_2:26</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_2_1_9"><span class="kw">suppose </span></a><a NAME="E1:42_2_1_9"/><span class="lab"><font color="Green" title="E34">A41</font></span>: 
( <font color="Maroon" title="c13">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="c12">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"><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#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c11">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"><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#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><span class="kw">then </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_2_1_9"><span class="lab"><font color="Green" title="E34">A41</font></span></a>, <a class="ref" href="waybel_1.html#T4" title="WAYBEL_1:th.4">WAYBEL_1:4</a></span>
;<br/><span class="kw">hence </span><a NAME="E3:42_2_1_9"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E8:42"/><span class="lab"><font color="Green" title="E32">A42</font></span>: 
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="lattice5.html#K6" title="LATTICE5:func.6">new_set</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="c4">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">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="42_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">p</font>, <font color="Maroon" title="c12">q</font>, <font color="Maroon" title="c13">u</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice5.html#K6" title="LATTICE5:func.6">new_set</a> <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> &amp; <font color="Maroon" title="c12">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">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> implies <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:42_3"/><span class="lab"><font color="Green" title="E33">A43</font></span>: 
<font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font>
 <span class="kw">and </span><br/><a NAME="E2:42_3"/><span class="lab"><font color="Green" title="E34">A44</font></span>: 
( <font color="Maroon" title="c12">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">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c14">q9</font> = <font color="Maroon" title="c12">q</font>, <font color="Maroon" title="c15">u9</font> = <font color="Maroon" title="c13">u</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42_3"><span class="lab"><font color="Green" title="E34">A44</font></span></a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_3_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:42_3_1"/>
( ( <font color="Maroon" title="c11">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="c12">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">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="c11">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="c12">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">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="c11">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"><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#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c12">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">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_3"><span class="lab"><font color="Green" title="E33">A43</font></span></a>, <a class="txt" href="#E2:42_3"><span class="lab"><font color="Green" title="E34">A44</font></span></a>, <a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:42_3_1_1"/><span class="lab"><font color="Green" title="E35">A45</font></span>: 
( <font color="Maroon" title="c11">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="c12">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">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><a NAME="E2:42_3_1_1"/>
<font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c15">u9</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">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="c15">u9</font>,<font color="Maroon" title="c14">q9</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="c14">q9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="ref" href="lattice5.html#D8" target="_self" title="LATTICE5:def.8">Def8</a></span>;<br/><a NAME="E3:42_3_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c15">u9</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">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="c14">q9</font>,<font color="Maroon" title="c15">u9</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="c14">q9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="ref" href="lattice5.html#D6" target="_self" title="LATTICE5:def.6">Def6</a></span>;<br/><a NAME="E4:42_3_1_1"/><span class="kw">then </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="c15">u9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c9">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="c14">q9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c14">q9</font>,<font color="Maroon" title="c15">u9</font></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_1.html#T3" title="WAYBEL_1:th.3">WAYBEL_1:3</a></span>;<br/><a NAME="E5:42_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A46</font></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="c15">u9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c9">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="c14">q9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c9">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="c14">q9</font>,<font color="Maroon" title="c15">u9</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></span>;<br/><a NAME="E6:42_3_1_1"/><span class="lab"><font color="Green" title="E37">A47</font></span>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">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="c14">q9</font>,<font color="Maroon" title="c15">u9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>;<br/><a NAME="E7:42_3_1_1"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="c14">q9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_3_1_1"><span class="lab"><font color="Green" title="E35">A45</font></span></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>;<br/><span class="kw">hence </span><a NAME="E8:42_3_1_1"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_3_1_1"><span class="lab"><font color="Green" title="E35">A45</font></span></a>, <a class="txt" href="#E6:42_3_1_1"><span class="lab"><font color="Green" title="E37">A47</font></span></a>, <a class="txt" href="#E5:42_3_1_1"><span class="lab"><font color="Green" title="E36">A46</font></span></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:42_3_1_2"/><span class="lab"><font color="Green" title="E35">A48</font></span>: 
( <font color="Maroon" title="c11">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="c12">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">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><a NAME="E2:42_3_1_2"/>
<font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c15">u9</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">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="c15">u9</font>,<font color="Maroon" title="c14">q9</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="c14">q9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="ref" href="lattice5.html#D8" target="_self" title="LATTICE5:def.8">Def8</a></span>;<br/><a NAME="E3:42_3_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c15">u9</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">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="c14">q9</font>,<font color="Maroon" title="c15">u9</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="c14">q9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="ref" href="lattice5.html#D6" target="_self" title="LATTICE5:def.6">Def6</a></span>;<br/><a NAME="E4:42_3_1_2"/><span class="kw">then </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="c15">u9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</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> <font color="Maroon" title="c14">q9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c14">q9</font>,<font color="Maroon" title="c15">u9</font></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_1.html#T3" title="WAYBEL_1:th.3">WAYBEL_1:3</a></span>;<br/><a NAME="E5:42_3_1_2"/><span class="kw">then </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="c15">u9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</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="c14">q9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c14">q9</font>,<font color="Maroon" title="c15">u9</font></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></span>;<br/><a NAME="E6:42_3_1_2"/><span class="kw">then </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="c15">u9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</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="c14">q9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></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="c14">q9</font>,<font color="Maroon" title="c15">u9</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></span>;<br/><a NAME="E7:42_3_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A49</font></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="c15">u9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</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"><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="c14">q9</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</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="c14">q9</font>,<font color="Maroon" title="c15">u9</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></span>;<br/><a NAME="E8:42_3_1_2"/><span class="lab"><font color="Green" title="E37">A50</font></span>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">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="c14">q9</font>,<font color="Maroon" title="c15">u9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>;<br/><a NAME="E9:42_3_1_2"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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> <font color="Maroon" title="c14">q9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_3_1_2"><span class="lab"><font color="Green" title="E35">A48</font></span></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>;<br/><span class="kw">hence </span><a NAME="E10:42_3_1_2"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_3_1_2"><span class="lab"><font color="Green" title="E35">A48</font></span></a>, <a class="txt" href="#E8:42_3_1_2"><span class="lab"><font color="Green" title="E37">A50</font></span></a>, <a class="txt" href="#E7:42_3_1_2"><span class="lab"><font color="Green" title="E36">A49</font></span></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_3_1_3"><span class="kw">suppose </span></a><a NAME="E1:42_3_1_3"/><span class="lab"><font color="Green" title="E35">A51</font></span>: 
( <font color="Maroon" title="c11">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"><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#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c12">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">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><a NAME="E2:42_3_1_3"/>
<font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c15">u9</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">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="c15">u9</font>,<font color="Maroon" title="c14">q9</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="c14">q9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="ref" href="lattice5.html#D8" target="_self" title="LATTICE5:def.8">Def8</a></span>;<br/><a NAME="E3:42_3_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c15">u9</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">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="c14">q9</font>,<font color="Maroon" title="c15">u9</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="c14">q9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="ref" href="lattice5.html#D6" target="_self" title="LATTICE5:def.6">Def6</a></span>;<br/><a NAME="E4:42_3_1_3"/><span class="kw">then </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="c15">u9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c10">b</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="c14">q9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c14">q9</font>,<font color="Maroon" title="c15">u9</font></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_1.html#T3" title="WAYBEL_1:th.3">WAYBEL_1:3</a></span>;<br/><a NAME="E5:42_3_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A52</font></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="c15">u9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c10">b</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="c14">q9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c10">b</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="c14">q9</font>,<font color="Maroon" title="c15">u9</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></span>;<br/><a NAME="E6:42_3_1_3"/><span class="lab"><font color="Green" title="E37">A53</font></span>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">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="c14">q9</font>,<font color="Maroon" title="c15">u9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>;<br/><a NAME="E7:42_3_1_3"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="c14">q9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c10">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_3_1_3"><span class="lab"><font color="Green" title="E35">A51</font></span></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>;<br/><span class="kw">hence </span><a NAME="E8:42_3_1_3"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_3_1_3"><span class="lab"><font color="Green" title="E35">A51</font></span></a>, <a class="txt" href="#E6:42_3_1_3"><span class="lab"><font color="Green" title="E37">A53</font></span></a>, <a class="txt" href="#E5:42_3_1_3"><span class="lab"><font color="Green" title="E36">A52</font></span></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E9:42"/><span class="lab"><font color="Green" title="E33">A54</font></span>: 
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="lattice5.html#K6" title="LATTICE5:func.6">new_set</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="c4">B</font> holds <br/><span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">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="42_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">p</font>, <font color="Maroon" title="c12">q</font>, <font color="Maroon" title="c13">u</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice5.html#K6" title="LATTICE5:func.6">new_set</a> <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">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="c12">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">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> implies <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:42_4"/><span class="lab"><font color="Green" title="E34">A55</font></span>: 
( <font color="Maroon" title="c11">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="c12">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">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_4_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:42_4_1"/>
( ( <font color="Maroon" title="c11">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="c12">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">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="c11">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="c12">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">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="c11">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="c12">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">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"><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#K1" title="TARSKI:func.1">}</a></span> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_4"><span class="lab"><font color="Green" title="E34">A55</font></span></a>, <a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_4_1_1"><span class="kw">suppose </span></a><a NAME="E1:42_4_1_1"/><span class="lab"><font color="Green" title="E35">A56</font></span>: 
( <font color="Maroon" title="c11">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="c12">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">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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">p9</font> = <font color="Maroon" title="c11">p</font>, <font color="Maroon" title="c15">q9</font> = <font color="Maroon" title="c12">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="E3:42_4_1_1"/><span class="lab"><font color="Green" title="E36">A57</font></span>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="c14">p9</font>,<font color="Maroon" title="c15">q9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>;<br/><a NAME="E4:42_4_1_1"/>
<font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p9</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">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="c14">p9</font>,<font color="Maroon" title="c15">q9</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="c15">q9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="ref" href="lattice5.html#D8" target="_self" title="LATTICE5:def.8">Def8</a></span>;<br/><a NAME="E5:42_4_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A58</font></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="c14">p9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c9">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="c14">p9</font>,<font color="Maroon" title="c15">q9</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="c15">q9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_1.html#T3" title="WAYBEL_1:th.3">WAYBEL_1:3</a></span>;<br/><a NAME="E6:42_4_1_1"/>
( <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="c14">p9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font> &amp; <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">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="c15">q9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_4_1_1"><span class="lab"><font color="Green" title="E35">A56</font></span></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>;<br/><span class="kw">hence </span><a NAME="E7:42_4_1_1"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42_4_1_1"><span class="lab"><font color="Green" title="E36">A57</font></span></a>, <a class="txt" href="#E5:42_4_1_1"><span class="lab"><font color="Green" title="E37">A58</font></span></a>, <a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_4_1_2"><span class="kw">suppose </span></a><a NAME="E1:42_4_1_2"/><span class="lab"><font color="Green" title="E35">A59</font></span>: 
( <font color="Maroon" title="c11">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="c12">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">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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">p9</font> = <font color="Maroon" title="c11">p</font>, <font color="Maroon" title="c15">q9</font> = <font color="Maroon" title="c12">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="E3:42_4_1_2"/><span class="lab"><font color="Green" title="E36">A60</font></span>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="c14">p9</font>,<font color="Maroon" title="c15">q9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>;<br/><a NAME="E4:42_4_1_2"/>
<font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p9</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">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="c14">p9</font>,<font color="Maroon" title="c15">q9</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="c15">q9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="ref" href="lattice5.html#D8" target="_self" title="LATTICE5:def.8">Def8</a></span>;<br/><a NAME="E5:42_4_1_2"/><span class="kw">then </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="c14">p9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c9">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="c14">p9</font>,<font color="Maroon" title="c15">q9</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="c15">q9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_1.html#T3" title="WAYBEL_1:th.3">WAYBEL_1:3</a></span>;<br/><a NAME="E6:42_4_1_2"/><span class="kw">then </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="c14">p9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</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"><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="c14">p9</font>,<font color="Maroon" title="c15">q9</font></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="c15">q9</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_1.html#T3" title="WAYBEL_1:th.3">WAYBEL_1:3</a></span>;<br/><a NAME="E7:42_4_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A61</font></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="c14">p9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</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="c14">p9</font>,<font color="Maroon" title="c15">q9</font></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="c15">q9</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></span>;<br/><a NAME="E8:42_4_1_2"/>
( <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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> <font color="Maroon" title="c14">p9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font> &amp; <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">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> <font color="Maroon" title="c15">q9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_4_1_2"><span class="lab"><font color="Green" title="E35">A59</font></span></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>;<br/><span class="kw">hence </span><a NAME="E9:42_4_1_2"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42_4_1_2"><span class="lab"><font color="Green" title="E36">A60</font></span></a>, <a class="txt" href="#E7:42_4_1_2"><span class="lab"><font color="Green" title="E37">A61</font></span></a>, <a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_4_1_3"><span class="kw">suppose </span></a><a NAME="E1:42_4_1_3"/><span class="lab"><font color="Green" title="E35">A62</font></span>: 
( <font color="Maroon" title="c11">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="c12">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">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"><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#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">p9</font> = <font color="Maroon" title="c11">p</font>, <font color="Maroon" title="c15">q9</font> = <font color="Maroon" title="c12">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="E3:42_4_1_3"/><span class="lab"><font color="Green" title="E36">A63</font></span>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="c14">p9</font>,<font color="Maroon" title="c15">q9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>;<br/><a NAME="E4:42_4_1_3"/>
<font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p9</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">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="c14">p9</font>,<font color="Maroon" title="c15">q9</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="c15">q9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="ref" href="lattice5.html#D8" target="_self" title="LATTICE5:def.8">Def8</a></span>;<br/><a NAME="E5:42_4_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A64</font></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="c14">p9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c10">b</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="c14">p9</font>,<font color="Maroon" title="c15">q9</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="c15">q9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c10">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_1.html#T3" title="WAYBEL_1:th.3">WAYBEL_1:3</a></span>;<br/><a NAME="E6:42_4_1_3"/>
( <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="c14">p9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c10">b</font> &amp; <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">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="c15">q9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c10">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_4_1_3"><span class="lab"><font color="Green" title="E35">A62</font></span></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>;<br/><span class="kw">hence </span><a NAME="E7:42_4_1_3"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42_4_1_3"><span class="lab"><font color="Green" title="E36">A63</font></span></a>, <a class="txt" href="#E5:42_4_1_3"><span class="lab"><font color="Green" title="E37">A64</font></span></a>, <a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E10:42"/><span class="lab"><font color="Green" title="E34">A65</font></span>: 
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="lattice5.html#K6" title="LATTICE5:func.6">new_set</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="c4">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="c4">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="c4">B</font> holds <br/><span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">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="42_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">p</font>, <font color="Maroon" title="c12">q</font>, <font color="Maroon" title="c13">u</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice5.html#K6" title="LATTICE5:func.6">new_set</a> <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> implies <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:42_5"/><span class="lab"><font color="Green" title="E35">A66</font></span>: 
( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_5_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:42_5_1"/>
( ( <font color="Maroon" title="c11">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="c12">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">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="c11">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="c12">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">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="c11">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="c12">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">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"><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#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c11">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="c12">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">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="c11">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="c12">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">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="c11">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="c12">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">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"><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#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c11">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="c12">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"><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#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">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="c11">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="c12">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"><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#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">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="c11">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="c12">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"><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#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">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"><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#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c11">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="c12">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">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="c11">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="c12">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">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="c11">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="c12">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">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"><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#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c11">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="c12">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">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="c11">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="c12">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">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="c11">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="c12">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">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"><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#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c11">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="c12">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"><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#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">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="c11">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="c12">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"><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#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">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="c11">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="c12">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"><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#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">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"><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#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c11">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"><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#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c12">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">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="c11">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"><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#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c12">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">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="c11">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"><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#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c12">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">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"><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#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c11">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"><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#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c12">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">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="c11">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"><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#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c12">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">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="c11">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"><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#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c12">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">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"><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#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c11">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"><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#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c12">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"><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#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">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="c11">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"><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#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c12">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"><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#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">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="c11">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"><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#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c12">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"><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#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">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"><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#K1" title="TARSKI:func.1">}</a></span> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_5"><span class="lab"><font color="Green" title="E35">A66</font></span></a>, <a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_5_1_1"><span class="kw">suppose </span></a><a NAME="E1:42_5_1_1"/><span class="lab"><font color="Green" title="E36">A67</font></span>: 
( <font color="Maroon" title="c11">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="c12">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">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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><a NAME="E2:42_5_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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T44" title="YELLOW_0:th.44">YELLOW_0:44</a></span>;<br/><span class="kw">hence </span><a NAME="E3:42_5_1_1"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_5_1_1"><span class="lab"><font color="Green" title="E36">A67</font></span></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_5_1_2"><span class="kw">suppose </span></a><a NAME="E1:42_5_1_2"/><span class="lab"><font color="Green" title="E36">A68</font></span>: 
( <font color="Maroon" title="c11">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="c12">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">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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><span class="kw">then </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>
<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> <font color="Maroon" title="c10">b</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_5_1_2"><span class="lab"><font color="Green" title="E36">A68</font></span></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>
<br/>.= 
<font color="Maroon" title="c10">b</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_1.html#T4" title="WAYBEL_1:th.4">WAYBEL_1:4</a></span>
;<br/><span class="kw">hence </span><a NAME="E3:42_5_1_2"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_5_1_2"><span class="lab"><font color="Green" title="E36">A68</font></span></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_5_1_3"><span class="kw">suppose </span></a><a NAME="E1:42_5_1_3"/><span class="lab"><font color="Green" title="E36">A69</font></span>: 
( <font color="Maroon" title="c11">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="c12">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">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"><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#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><span class="kw">then </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>
<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"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_5_1_3"><span class="lab"><font color="Green" title="E36">A69</font></span></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>
<br/>.= 
<font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_1.html#T4" title="WAYBEL_1:th.4">WAYBEL_1:4</a></span>
;<br/><span class="kw">hence </span><a NAME="E3:42_5_1_3"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_5_1_3"><span class="lab"><font color="Green" title="E36">A69</font></span></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_5_1_4"><span class="kw">suppose </span></a><a NAME="E1:42_5_1_4"/><span class="lab"><font color="Green" title="E36">A70</font></span>: 
( <font color="Maroon" title="c11">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="c12">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">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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><a NAME="E2:42_5_1_4"/>
 <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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T44" title="YELLOW_0:th.44">YELLOW_0:44</a></span>;<br/><span class="kw">hence </span><a NAME="E3:42_5_1_4"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_5_1_4"><span class="lab"><font color="Green" title="E36">A70</font></span></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_5_1_5"><span class="kw">suppose </span></a><a NAME="E1:42_5_1_5"/><span class="lab"><font color="Green" title="E36">A71</font></span>: 
( <font color="Maroon" title="c11">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="c12">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">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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><span class="kw">then </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span> = 
<font color="Maroon" title="c10">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>
<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> <font color="Maroon" title="c10">b</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_5_1_5"><span class="lab"><font color="Green" title="E36">A71</font></span></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>
<br/>.= 
<font color="Maroon" title="c10">b</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_1.html#T4" title="WAYBEL_1:th.4">WAYBEL_1:4</a></span>
;<br/><span class="kw">hence </span><a NAME="E3:42_5_1_5"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_5_1_5"><span class="lab"><font color="Green" title="E36">A71</font></span></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_5_1_6"><span class="kw">suppose </span></a><a NAME="E1:42_5_1_6"/><span class="lab"><font color="Green" title="E36">A72</font></span>: 
( <font color="Maroon" title="c11">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="c12">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">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"><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#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><span class="kw">then </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span> = 
<font color="Maroon" title="c10">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>
<br/>.= 
<font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_5_1_6"><span class="lab"><font color="Green" title="E36">A72</font></span></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>
;<br/><span class="kw">hence </span><a NAME="E3:42_5_1_6"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_5_1_6"><span class="lab"><font color="Green" title="E36">A72</font></span></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_5_1_7"><span class="kw">suppose </span></a><a NAME="E1:42_5_1_7"/><span class="lab"><font color="Green" title="E36">A73</font></span>: 
( <font color="Maroon" title="c11">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="c12">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"><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#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><a NAME="E2:42_5_1_7"/>
 <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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T44" title="YELLOW_0:th.44">YELLOW_0:44</a></span>;<br/><span class="kw">hence </span><a NAME="E3:42_5_1_7"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_5_1_7"><span class="lab"><font color="Green" title="E36">A73</font></span></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_5_1_8"><span class="kw">suppose </span></a><a NAME="E1:42_5_1_8"/><span class="lab"><font color="Green" title="E36">A74</font></span>: 
( <font color="Maroon" title="c11">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="c12">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"><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#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><a NAME="E2:42_5_1_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A75</font></span>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_5_1_8"><span class="lab"><font color="Green" title="E36">A74</font></span></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_5_1_8"><span class="lab"><font color="Green" title="E36">A74</font></span></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>
<br/>.= 
<font color="Maroon" title="c10">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></span>
<br/>.= 
<font color="Maroon" title="c10">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_5.html#T1" title="YELLOW_5:th.1">YELLOW_5:1</a></span>
;<br/><span class="kw">hence </span><a NAME="E4:42_5_1_8"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42_5_1_8"><span class="lab"><font color="Green" title="E37">A75</font></span></a>, <a class="ref" href="yellow_0.html#T22" title="YELLOW_0:th.22">YELLOW_0:22</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_5_1_9"><span class="kw">suppose </span></a><a NAME="E1:42_5_1_9"/><span class="lab"><font color="Green" title="E36">A76</font></span>: 
( <font color="Maroon" title="c11">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="c12">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"><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#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">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"><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#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><span class="kw">then </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>
<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"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_5_1_9"><span class="lab"><font color="Green" title="E36">A76</font></span></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>
<br/>.= 
<font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_1.html#T4" title="WAYBEL_1:th.4">WAYBEL_1:4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_5_1_9"><span class="lab"><font color="Green" title="E36">A76</font></span></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>
;<br/><span class="kw">hence </span><a NAME="E3:42_5_1_9"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_5_1_9"><span class="lab"><font color="Green" title="E36">A76</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_5_1_10"><span class="kw">suppose </span></a><a NAME="E1:42_5_1_10"/><span class="lab"><font color="Green" title="E36">A77</font></span>: 
( <font color="Maroon" title="c11">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="c12">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">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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><span class="kw">then </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span> = 
<font color="Maroon" title="c10">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>
<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> <font color="Maroon" title="c10">b</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_5_1_10"><span class="lab"><font color="Green" title="E36">A77</font></span></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>
<br/>.= 
<font color="Maroon" title="c10">b</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_1.html#T4" title="WAYBEL_1:th.4">WAYBEL_1:4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_5_1_10"><span class="lab"><font color="Green" title="E36">A77</font></span></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>
;<br/><span class="kw">hence </span><a NAME="E3:42_5_1_10"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_5_1_10"><span class="lab"><font color="Green" title="E36">A77</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_5_1_11"><span class="kw">suppose </span></a><a NAME="E1:42_5_1_11"/><span class="lab"><font color="Green" title="E36">A78</font></span>: 
( <font color="Maroon" title="c11">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="c12">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">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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><a NAME="E2:42_5_1_11"/>
 <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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T44" title="YELLOW_0:th.44">YELLOW_0:44</a></span>;<br/><span class="kw">hence </span><a NAME="E3:42_5_1_11"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_5_1_11"><span class="lab"><font color="Green" title="E36">A78</font></span></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_5_1_12"><span class="kw">suppose </span></a><a NAME="E1:42_5_1_12"/><span class="lab"><font color="Green" title="E36">A79</font></span>: 
( <font color="Maroon" title="c11">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="c12">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">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"><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#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><a NAME="E2:42_5_1_12"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A80</font></span>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span> = 
<font color="Maroon" title="c10">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_5_1_12"><span class="lab"><font color="Green" title="E36">A79</font></span></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>
<br/>.= 
<font color="Maroon" title="c10">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_5_1_12"><span class="lab"><font color="Green" title="E36">A79</font></span></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></span>
<br/>.= 
<font color="Maroon" title="c10">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_5.html#T1" title="YELLOW_5:th.1">YELLOW_5:1</a></span>
;<br/><span class="kw">hence </span><a NAME="E4:42_5_1_12"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42_5_1_12"><span class="lab"><font color="Green" title="E37">A80</font></span></a>, <a class="ref" href="yellow_0.html#T22" title="YELLOW_0:th.22">YELLOW_0:22</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_5_1_13"><span class="kw">suppose </span></a><a NAME="E1:42_5_1_13"/><span class="lab"><font color="Green" title="E36">A81</font></span>: 
( <font color="Maroon" title="c11">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="c12">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">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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><span class="kw">then </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>
<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> <font color="Maroon" title="c10">b</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_5_1_13"><span class="lab"><font color="Green" title="E36">A81</font></span></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>
<br/>.= 
<font color="Maroon" title="c10">b</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_1.html#T4" title="WAYBEL_1:th.4">WAYBEL_1:4</a></span>
;<br/><span class="kw">hence </span><a NAME="E3:42_5_1_13"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_5_1_13"><span class="lab"><font color="Green" title="E36">A81</font></span></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_5_1_14"><span class="kw">suppose </span></a><a NAME="E1:42_5_1_14"/><span class="lab"><font color="Green" title="E36">A82</font></span>: 
( <font color="Maroon" title="c11">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="c12">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">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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><a NAME="E2:42_5_1_14"/>
 <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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T44" title="YELLOW_0:th.44">YELLOW_0:44</a></span>;<br/><span class="kw">hence </span><a NAME="E3:42_5_1_14"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_5_1_14"><span class="lab"><font color="Green" title="E36">A82</font></span></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_5_1_15"><span class="kw">suppose </span></a><a NAME="E1:42_5_1_15"/><span class="lab"><font color="Green" title="E36">A83</font></span>: 
( <font color="Maroon" title="c11">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="c12">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">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"><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#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><span class="kw">then </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>
<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> <font color="Maroon" title="c9">a</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_5_1_15"><span class="lab"><font color="Green" title="E36">A83</font></span></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>
<br/>.= 
<font color="Maroon" title="c9">a</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_1.html#T4" title="WAYBEL_1:th.4">WAYBEL_1:4</a></span>
;<br/><span class="kw">hence </span><a NAME="E3:42_5_1_15"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_5_1_15"><span class="lab"><font color="Green" title="E36">A83</font></span></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_5_1_16"><span class="kw">suppose </span></a><a NAME="E1:42_5_1_16"/><span class="lab"><font color="Green" title="E36">A84</font></span>: 
( <font color="Maroon" title="c11">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="c12">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"><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#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><a NAME="E2:42_5_1_16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A85</font></span>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span> = 
<font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_5_1_16"><span class="lab"><font color="Green" title="E36">A84</font></span></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>
<br/>.= 
<font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_5_1_16"><span class="lab"><font color="Green" title="E36">A84</font></span></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></span>
<br/>.= 
<font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_5.html#T1" title="YELLOW_5:th.1">YELLOW_5:1</a></span>
;<br/><span class="kw">hence </span><a NAME="E4:42_5_1_16"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42_5_1_16"><span class="lab"><font color="Green" title="E37">A85</font></span></a>, <a class="ref" href="yellow_0.html#T22" title="YELLOW_0:th.22">YELLOW_0:22</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_5_1_17"><span class="kw">suppose </span></a><a NAME="E1:42_5_1_17"/><span class="lab"><font color="Green" title="E36">A86</font></span>: 
( <font color="Maroon" title="c11">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="c12">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"><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#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><a NAME="E2:42_5_1_17"/>
 <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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T44" title="YELLOW_0:th.44">YELLOW_0:44</a></span>;<br/><span class="kw">hence </span><a NAME="E3:42_5_1_17"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_5_1_17"><span class="lab"><font color="Green" title="E36">A86</font></span></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_5_1_18"><span class="kw">suppose </span></a><a NAME="E1:42_5_1_18"/><span class="lab"><font color="Green" title="E36">A87</font></span>: 
( <font color="Maroon" title="c11">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="c12">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"><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#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">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"><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#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><span class="kw">then </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span> = 
<font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>
<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> <font color="Maroon" title="c9">a</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_5_1_18"><span class="lab"><font color="Green" title="E36">A87</font></span></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>
<br/>.= 
<font color="Maroon" title="c9">a</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_1.html#T4" title="WAYBEL_1:th.4">WAYBEL_1:4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_5_1_18"><span class="lab"><font color="Green" title="E36">A87</font></span></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>
;<br/><span class="kw">hence </span><a NAME="E3:42_5_1_18"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_5_1_18"><span class="lab"><font color="Green" title="E36">A87</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_5_1_19"><span class="kw">suppose </span></a><a NAME="E1:42_5_1_19"/><span class="lab"><font color="Green" title="E36">A88</font></span>: 
( <font color="Maroon" title="c11">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"><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#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c12">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">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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><span class="kw">then </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>
<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"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_5_1_19"><span class="lab"><font color="Green" title="E36">A88</font></span></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>
<br/>.= 
<font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_1.html#T4" title="WAYBEL_1:th.4">WAYBEL_1:4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_5_1_19"><span class="lab"><font color="Green" title="E36">A88</font></span></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>
;<br/><span class="kw">hence </span><a NAME="E3:42_5_1_19"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_5_1_19"><span class="lab"><font color="Green" title="E36">A88</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_5_1_20"><span class="kw">suppose </span></a><a NAME="E1:42_5_1_20"/><span class="lab"><font color="Green" title="E36">A89</font></span>: 
( <font color="Maroon" title="c11">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"><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#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c12">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">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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><a NAME="E2:42_5_1_20"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A90</font></span>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_5_1_20"><span class="lab"><font color="Green" title="E36">A89</font></span></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_5_1_20"><span class="lab"><font color="Green" title="E36">A89</font></span></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>
<br/>.= 
<font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></span>
<br/>.= 
<font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_5.html#T1" title="YELLOW_5:th.1">YELLOW_5:1</a></span>
;<br/><span class="kw">hence </span><a NAME="E4:42_5_1_20"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42_5_1_20"><span class="lab"><font color="Green" title="E37">A90</font></span></a>, <a class="ref" href="yellow_0.html#T22" title="YELLOW_0:th.22">YELLOW_0:22</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_5_1_21"><span class="kw">suppose </span></a><a NAME="E1:42_5_1_21"/><span class="lab"><font color="Green" title="E36">A91</font></span>: 
( <font color="Maroon" title="c11">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"><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#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c12">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">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"><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#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><a NAME="E2:42_5_1_21"/>
 <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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T44" title="YELLOW_0:th.44">YELLOW_0:44</a></span>;<br/><span class="kw">hence </span><a NAME="E3:42_5_1_21"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_5_1_21"><span class="lab"><font color="Green" title="E36">A91</font></span></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_5_1_22"><span class="kw">suppose </span></a><a NAME="E1:42_5_1_22"/><span class="lab"><font color="Green" title="E36">A92</font></span>: 
( <font color="Maroon" title="c11">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"><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#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c12">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">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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><span class="kw">then </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span> = 
<font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>
<br/>.= 
<font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_5_1_22"><span class="lab"><font color="Green" title="E36">A92</font></span></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>
;<br/><span class="kw">hence </span><a NAME="E3:42_5_1_22"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_5_1_22"><span class="lab"><font color="Green" title="E36">A92</font></span></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_5_1_23"><span class="kw">suppose </span></a><a NAME="E1:42_5_1_23"/><span class="lab"><font color="Green" title="E36">A93</font></span>: 
( <font color="Maroon" title="c11">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"><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#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c12">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">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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><span class="kw">then </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span> = 
<font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>
<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> <font color="Maroon" title="c9">a</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_5_1_23"><span class="lab"><font color="Green" title="E36">A93</font></span></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>
<br/>.= 
<font color="Maroon" title="c9">a</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_1.html#T4" title="WAYBEL_1:th.4">WAYBEL_1:4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_5_1_23"><span class="lab"><font color="Green" title="E36">A93</font></span></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>
;<br/><span class="kw">hence </span><a NAME="E3:42_5_1_23"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_5_1_23"><span class="lab"><font color="Green" title="E36">A93</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_5_1_24"><span class="kw">suppose </span></a><a NAME="E1:42_5_1_24"/><span class="lab"><font color="Green" title="E36">A94</font></span>: 
( <font color="Maroon" title="c11">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"><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#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c12">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">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"><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#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><a NAME="E2:42_5_1_24"/>
 <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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T44" title="YELLOW_0:th.44">YELLOW_0:44</a></span>;<br/><span class="kw">hence </span><a NAME="E3:42_5_1_24"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_5_1_24"><span class="lab"><font color="Green" title="E36">A94</font></span></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_5_1_25"><span class="kw">suppose </span></a><a NAME="E1:42_5_1_25"/><span class="lab"><font color="Green" title="E36">A95</font></span>: 
( <font color="Maroon" title="c11">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"><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#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c12">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"><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#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><span class="kw">then </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>
<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"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_5_1_25"><span class="lab"><font color="Green" title="E36">A95</font></span></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>
<br/>.= 
<font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_1.html#T4" title="WAYBEL_1:th.4">WAYBEL_1:4</a></span>
;<br/><span class="kw">hence </span><a NAME="E3:42_5_1_25"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_5_1_25"><span class="lab"><font color="Green" title="E36">A95</font></span></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_5_1_26"><span class="kw">suppose </span></a><a NAME="E1:42_5_1_26"/><span class="lab"><font color="Green" title="E36">A96</font></span>: 
( <font color="Maroon" title="c11">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"><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#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c12">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"><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#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><span class="kw">then </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>
<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> <font color="Maroon" title="c9">a</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_5_1_26"><span class="lab"><font color="Green" title="E36">A96</font></span></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>
<br/>.= 
<font color="Maroon" title="c9">a</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_1.html#T4" title="WAYBEL_1:th.4">WAYBEL_1:4</a></span>
;<br/><span class="kw">hence </span><a NAME="E3:42_5_1_26"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_5_1_26"><span class="lab"><font color="Green" title="E36">A96</font></span></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_5_1_27"><span class="kw">suppose </span></a><a NAME="E1:42_5_1_27"/><span class="lab"><font color="Green" title="E36">A97</font></span>: 
( <font color="Maroon" title="c11">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"><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#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c12">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"><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#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">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"><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#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><a NAME="E2:42_5_1_27"/>
 <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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T44" title="YELLOW_0:th.44">YELLOW_0:44</a></span>;<br/><span class="kw">hence </span><a NAME="E3:42_5_1_27"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_5_1_27"><span class="lab"><font color="Green" title="E36">A97</font></span></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E11:42"/><span class="lab"><font color="Green" title="E35">A98</font></span>: 
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="lattice5.html#K6" title="LATTICE5:func.6">new_set</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="c4">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="c4">B</font> holds <br/><span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">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="42_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">p</font>, <font color="Maroon" title="c12">q</font>, <font color="Maroon" title="c13">u</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice5.html#K6" title="LATTICE5:func.6">new_set</a> <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> &amp; <font color="Maroon" title="c12">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">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> implies <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:42_6"/><span class="lab"><font color="Green" title="E36">A99</font></span>: 
<font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font>
 <span class="kw">and </span><br/><a NAME="E2:42_6"/><span class="lab"><font color="Green" title="E37">A100</font></span>: 
<font color="Maroon" title="c12">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">and </span><br/><a NAME="E3:42_6"/><span class="lab"><font color="Green" title="E38">A101</font></span>: 
<font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c14">q9</font> = <font color="Maroon" title="c12">q</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42_6"><span class="lab"><font color="Green" title="E37">A100</font></span></a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_6_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:42_6_1"/>
( ( <font color="Maroon" title="c12">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="c11">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="c13">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="c12">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="c11">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="c13">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="c12">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="c11">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="c13">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"><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#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c12">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="c11">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="c13">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="c12">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="c11">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="c13">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="c12">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="c11">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="c13">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"><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#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c12">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="c11">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"><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#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">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="c12">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="c11">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"><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#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">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="c12">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="c11">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"><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#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">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"><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#K1" title="TARSKI:func.1">}</a></span> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_6"><span class="lab"><font color="Green" title="E36">A99</font></span></a>, <a class="txt" href="#E2:42_6"><span class="lab"><font color="Green" title="E37">A100</font></span></a>, <a class="txt" href="#E3:42_6"><span class="lab"><font color="Green" title="E38">A101</font></span></a>, <a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_6_1_1"><span class="kw">suppose </span></a><a NAME="E1:42_6_1_1"/>
( <font color="Maroon" title="c12">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="c11">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="c13">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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><a NAME="E2:42_6_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>;<br/><span class="kw">hence </span><a NAME="E3:42_6_1_1"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T44" title="YELLOW_0:th.44">YELLOW_0:44</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_6_1_2"><span class="kw">suppose </span></a><a NAME="E1:42_6_1_2"/><span class="lab"><font color="Green" title="E39">A102</font></span>: 
( <font color="Maroon" title="c12">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="c11">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="c13">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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A103</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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"><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="c14">q9</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="c14">q9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></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="c14">q9</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></span>
;<br/><a NAME="E3:42_6_1_2"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_6_1_2"><span class="lab"><font color="Green" title="E39">A102</font></span></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>;<br/><span class="kw">hence </span><a NAME="E4:42_6_1_2"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42_6_1_2"><span class="lab"><font color="Green" title="E40">A103</font></span></a>, <a class="ref" href="yellow_0.html#T22" title="YELLOW_0:th.22">YELLOW_0:22</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_6_1_3"><span class="kw">suppose </span></a><a NAME="E1:42_6_1_3"/><span class="lab"><font color="Green" title="E39">A104</font></span>: 
( <font color="Maroon" title="c12">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="c11">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="c13">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"><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#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><a NAME="E2:42_6_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A105</font></span>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">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="c14">q9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c9">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"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_6_1_3"><span class="lab"><font color="Green" title="E39">A104</font></span></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>
<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="c14">q9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c9">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="c14">q9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c10">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_6_1_3"><span class="lab"><font color="Green" title="E39">A104</font></span></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>
<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="c14">q9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font> <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="c14">q9</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c5">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="c10">b</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></span>
<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="c14">q9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c14">q9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></span>
<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="c14">q9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c14">q9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></span>
;<br/><span class="kw">hence </span><a NAME="E4:42_6_1_3"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42_6_1_3"><span class="lab"><font color="Green" title="E40">A105</font></span></a>, <a class="ref" href="yellow_0.html#T22" title="YELLOW_0:th.22">YELLOW_0:22</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_6_1_4"><span class="kw">suppose </span></a><a NAME="E1:42_6_1_4"/><span class="lab"><font color="Green" title="E39">A106</font></span>: 
( <font color="Maroon" title="c12">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="c11">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="c13">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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A107</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">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> <font color="Maroon" title="c14">q9</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</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="c14">q9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</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="c14">q9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></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="c14">q9</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></span>
;<br/><a NAME="E3:42_6_1_4"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_6_1_4"><span class="lab"><font color="Green" title="E39">A106</font></span></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>;<br/><span class="kw">hence </span><a NAME="E4:42_6_1_4"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42_6_1_4"><span class="lab"><font color="Green" title="E40">A107</font></span></a>, <a class="ref" href="yellow_0.html#T22" title="YELLOW_0:th.22">YELLOW_0:22</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_6_1_5"><span class="kw">suppose </span></a><a NAME="E1:42_6_1_5"/>
( <font color="Maroon" title="c12">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="c11">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="c13">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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><a NAME="E2:42_6_1_5"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>;<br/><span class="kw">hence </span><a NAME="E3:42_6_1_5"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T44" title="YELLOW_0:th.44">YELLOW_0:44</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_6_1_6"><span class="kw">suppose </span></a><a NAME="E1:42_6_1_6"/><span class="lab"><font color="Green" title="E39">A108</font></span>: 
( <font color="Maroon" title="c12">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="c11">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="c13">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"><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#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A109</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">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> <font color="Maroon" title="c14">q9</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <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="c14">q9</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c5">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="c10">b</font></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"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></span>
<br/>.= 
<font color="Maroon" title="c9">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> <font color="Maroon" title="c14">q9</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c5">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="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></span>
;<br/><a NAME="E3:42_6_1_6"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_6_1_6"><span class="lab"><font color="Green" title="E39">A108</font></span></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>;<br/><span class="kw">hence </span><a NAME="E4:42_6_1_6"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42_6_1_6"><span class="lab"><font color="Green" title="E40">A109</font></span></a>, <a class="ref" href="yellow_0.html#T22" title="YELLOW_0:th.22">YELLOW_0:22</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_6_1_7"><span class="kw">suppose </span></a><a NAME="E1:42_6_1_7"/><span class="lab"><font color="Green" title="E39">A110</font></span>: 
( <font color="Maroon" title="c12">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="c11">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"><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#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><a NAME="E2:42_6_1_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A111</font></span>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">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="c14">q9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c10">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_6_1_7"><span class="lab"><font color="Green" title="E39">A110</font></span></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>
<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="c14">q9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c10">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"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_6_1_7"><span class="lab"><font color="Green" title="E39">A110</font></span></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>
<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="c14">q9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c10">b</font> <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="c14">q9</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></span>
<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="c14">q9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c14">q9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c10">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></span>
<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="c14">q9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c14">q9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></span>
;<br/><span class="kw">hence </span><a NAME="E4:42_6_1_7"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42_6_1_7"><span class="lab"><font color="Green" title="E40">A111</font></span></a>, <a class="ref" href="yellow_0.html#T22" title="YELLOW_0:th.22">YELLOW_0:22</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_6_1_8"><span class="kw">suppose </span></a><a NAME="E1:42_6_1_8"/><span class="lab"><font color="Green" title="E39">A112</font></span>: 
( <font color="Maroon" title="c12">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="c11">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"><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#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A113</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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"><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="c14">q9</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="c14">q9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></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="c14">q9</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c5">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="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></span>
;<br/><a NAME="E3:42_6_1_8"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_6_1_8"><span class="lab"><font color="Green" title="E39">A112</font></span></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>;<br/><span class="kw">hence </span><a NAME="E4:42_6_1_8"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42_6_1_8"><span class="lab"><font color="Green" title="E40">A113</font></span></a>, <a class="ref" href="yellow_0.html#T22" title="YELLOW_0:th.22">YELLOW_0:22</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_6_1_9"><span class="kw">suppose </span></a><a NAME="E1:42_6_1_9"/>
( <font color="Maroon" title="c12">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="c11">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"><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#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">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"><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#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><a NAME="E2:42_6_1_9"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>;<br/><span class="kw">hence </span><a NAME="E3:42_6_1_9"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T44" title="YELLOW_0:th.44">YELLOW_0:44</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E12:42"/><span class="lab"><font color="Green" title="E36">A114</font></span>: 
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="lattice5.html#K6" title="LATTICE5:func.6">new_set</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="c4">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="c4">B</font> holds <br/><span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">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="42_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">p</font>, <font color="Maroon" title="c12">q</font>, <font color="Maroon" title="c13">u</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice5.html#K6" title="LATTICE5:func.6">new_set</a> <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">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="c12">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> implies <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:42_7"/><span class="lab"><font color="Green" title="E37">A115</font></span>: 
<font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">and </span><br/><a NAME="E2:42_7"/><span class="lab"><font color="Green" title="E38">A116</font></span>: 
( <font color="Maroon" title="c12">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c14">p9</font> = <font color="Maroon" title="c11">p</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_7"><span class="lab"><font color="Green" title="E37">A115</font></span></a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_7_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:42_7_1"/>
( ( <font color="Maroon" title="c11">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="c12">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">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="c11">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="c12">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">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="c11">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="c12">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">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"><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#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c11">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="c12">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">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="c11">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="c12">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">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="c11">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="c12">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">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"><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#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c11">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="c12">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"><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#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">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="c11">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="c12">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"><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#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">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="c11">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="c12">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"><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#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">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"><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#K1" title="TARSKI:func.1">}</a></span> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_7"><span class="lab"><font color="Green" title="E37">A115</font></span></a>, <a class="txt" href="#E2:42_7"><span class="lab"><font color="Green" title="E38">A116</font></span></a>, <a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_7_1_1"><span class="kw">suppose </span></a><a NAME="E1:42_7_1_1"/><span class="lab"><font color="Green" title="E39">A117</font></span>: 
( <font color="Maroon" title="c11">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="c12">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">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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><span class="kw">then </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_1.html#T4" title="WAYBEL_1:th.4">WAYBEL_1:4</a></span>
;<br/><span class="kw">hence </span><a NAME="E3:42_7_1_1"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_7_1_1"><span class="lab"><font color="Green" title="E39">A117</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_7_1_2"><span class="kw">suppose </span></a><a NAME="E1:42_7_1_2"/><span class="lab"><font color="Green" title="E39">A118</font></span>: 
( <font color="Maroon" title="c11">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="c12">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">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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><span class="kw">then </span><span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> = 
<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="c14">p9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_7_1_2"><span class="lab"><font color="Green" title="E39">A118</font></span></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>
;<br/><span class="kw">hence </span><a NAME="E3:42_7_1_2"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_7_1_2"><span class="lab"><font color="Green" title="E39">A118</font></span></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_7_1_3"><span class="kw">suppose </span></a><a NAME="E1:42_7_1_3"/><span class="lab"><font color="Green" title="E39">A119</font></span>: 
( <font color="Maroon" title="c11">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="c12">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">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"><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#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><a NAME="E2:42_7_1_3"/>
<font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p9</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">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="c14">p9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c5">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="c5">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="ref" href="lattice5.html#D8" target="_self" title="LATTICE5:def.8">Def8</a></span>;<br/><a NAME="E3:42_7_1_3"/><span class="kw">then </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="c14">p9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c10">b</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="c14">p9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c5">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="c5">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="c10">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_1.html#T3" title="WAYBEL_1:th.3">WAYBEL_1:3</a></span>;<br/><a NAME="E4:42_7_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A120</font></span>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="c14">p9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c5">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="c5">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="c10">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_7_1_3"><span class="lab"><font color="Green" title="E39">A119</font></span></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>;<br/><a NAME="E5:42_7_1_3"/>
<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="c14">p9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c10">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> <font color="Maroon" title="c14">p9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c10">b</font>
 ;<br/><a NAME="E6:42_7_1_3"/><span class="kw">then </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="c14">p9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c5">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="c5">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="c10">b</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> <font color="Maroon" title="c14">p9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c10">b</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> <span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c5">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> &amp; <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="c14">p9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c10">b</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> <span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c5">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="c14">p9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:42"><span class="lab"><font color="Green" title="E30">A14</font></span></a>, <a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a>, <a class="ref" href="yellow_3.html#T3" title="YELLOW_3:th.3">YELLOW_3:3</a></span>;<br/><a NAME="E7:42_7_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A121</font></span>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="c14">p9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:42_7_1_3"><span class="lab"><font color="Green" title="E40">A120</font></span></a>, <a class="ref" href="orders_2.html#T26" title="ORDERS_2:th.26">ORDERS_2:26</a></span>;<br/><span class="lab"><font color="Green" title="E42">A122</font></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="c14">p9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</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="c14">p9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c10">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></span>
<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="c14">p9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></span>
<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="c14">p9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_5.html#T1" title="YELLOW_5:th.1">YELLOW_5:1</a></span>
<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="c14">p9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_5.html#T1" title="YELLOW_5:th.1">YELLOW_5:1</a></span>
<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="c14">p9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></span>
<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="c14">p9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></span>
;<br/><a NAME="E9:42_7_1_3"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="c14">p9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_7_1_3"><span class="lab"><font color="Green" title="E39">A119</font></span></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>;<br/><span class="kw">hence </span><a NAME="E10:42_7_1_3"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_7_1_3"><span class="lab"><font color="Green" title="E39">A119</font></span></a>, <a class="txt" href="#E7:42_7_1_3"><span class="lab"><font color="Green" title="E41">A121</font></span></a>, <a class="txt" href="#E8:42_7_1_3"><span class="lab"><font color="Green" title="E42">A122</font></span></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_7_1_4"><span class="kw">suppose </span></a><a NAME="E1:42_7_1_4"/><span class="lab"><font color="Green" title="E39">A123</font></span>: 
( <font color="Maroon" title="c11">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="c12">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">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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><span class="kw">then </span><span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font> = 
<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="c14">p9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_7_1_4"><span class="lab"><font color="Green" title="E39">A123</font></span></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>
;<br/><span class="kw">then </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_7_1_4"><span class="lab"><font color="Green" title="E39">A123</font></span></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_5.html#T1" title="YELLOW_5:th.1">YELLOW_5:1</a></span>
;<br/><span class="kw">hence </span><a NAME="E4:42_7_1_4"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T22" title="YELLOW_0:th.22">YELLOW_0:22</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_7_1_5"><span class="kw">suppose </span></a><a NAME="E1:42_7_1_5"/><span class="lab"><font color="Green" title="E39">A124</font></span>: 
( <font color="Maroon" title="c11">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="c12">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">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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><span class="kw">then </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_1.html#T4" title="WAYBEL_1:th.4">WAYBEL_1:4</a></span>
;<br/><span class="kw">hence </span><a NAME="E3:42_7_1_5"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_7_1_5"><span class="lab"><font color="Green" title="E39">A124</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_7_1_6"><span class="kw">suppose </span></a><a NAME="E1:42_7_1_6"/><span class="lab"><font color="Green" title="E39">A125</font></span>: 
( <font color="Maroon" title="c11">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="c12">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">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"><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#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><a NAME="E2:42_7_1_6"/>
<font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p9</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">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="c14">p9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c5">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="c5">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="ref" href="lattice5.html#D8" target="_self" title="LATTICE5:def.8">Def8</a></span>;<br/><a NAME="E3:42_7_1_6"/><span class="kw">then </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="c14">p9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c10">b</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="c14">p9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c5">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="c5">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="c10">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_1.html#T3" title="WAYBEL_1:th.3">WAYBEL_1:3</a></span>;<br/><a NAME="E4:42_7_1_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A126</font></span>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="c14">p9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c5">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="c5">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="c10">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_7_1_6"><span class="lab"><font color="Green" title="E39">A125</font></span></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>;<br/><a NAME="E5:42_7_1_6"/>
<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="c14">p9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c10">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> <font color="Maroon" title="c14">p9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c10">b</font>
 ;<br/><a NAME="E6:42_7_1_6"/><span class="kw">then </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="c14">p9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c5">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="c5">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="c10">b</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> <font color="Maroon" title="c14">p9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c10">b</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> <span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c5">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> &amp; <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="c14">p9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c10">b</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> <span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c5">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="c14">p9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:42"><span class="lab"><font color="Green" title="E30">A14</font></span></a>, <a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a>, <a class="ref" href="yellow_3.html#T3" title="YELLOW_3:th.3">YELLOW_3:3</a></span>;<br/><a NAME="E7:42_7_1_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A127</font></span>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="c14">p9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:42_7_1_6"><span class="lab"><font color="Green" title="E40">A126</font></span></a>, <a class="ref" href="orders_2.html#T26" title="ORDERS_2:th.26">ORDERS_2:26</a></span>;<br/><span class="lab"><font color="Green" title="E42">A128</font></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="c14">p9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</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="c14">p9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c10">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></span>
<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="c14">p9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></span>
<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="c14">p9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_5.html#T1" title="YELLOW_5:th.1">YELLOW_5:1</a></span>
<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="c14">p9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_5.html#T1" title="YELLOW_5:th.1">YELLOW_5:1</a></span>
<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="c14">p9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></span>
<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="c14">p9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></span>
<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="c14">p9</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></span>
;<br/><a NAME="E9:42_7_1_6"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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> <font color="Maroon" title="c14">p9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_7_1_6"><span class="lab"><font color="Green" title="E39">A125</font></span></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>;<br/><span class="kw">hence </span><a NAME="E10:42_7_1_6"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_7_1_6"><span class="lab"><font color="Green" title="E39">A125</font></span></a>, <a class="txt" href="#E7:42_7_1_6"><span class="lab"><font color="Green" title="E41">A127</font></span></a>, <a class="txt" href="#E8:42_7_1_6"><span class="lab"><font color="Green" title="E42">A128</font></span></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_7_1_7"><span class="kw">suppose </span></a><a NAME="E1:42_7_1_7"/><span class="lab"><font color="Green" title="E39">A129</font></span>: 
( <font color="Maroon" title="c11">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="c12">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"><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#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><a NAME="E2:42_7_1_7"/>
<font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p9</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">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="c14">p9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c5">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="ref" href="lattice5.html#D8" target="_self" title="LATTICE5:def.8">Def8</a></span>;<br/><a NAME="E3:42_7_1_7"/><span class="kw">then </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="c14">p9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c9">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="c14">p9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c5">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_1.html#T3" title="WAYBEL_1:th.3">WAYBEL_1:3</a></span>;<br/><a NAME="E4:42_7_1_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A130</font></span>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="c14">p9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c5">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_7_1_7"><span class="lab"><font color="Green" title="E39">A129</font></span></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>;<br/><a NAME="E5:42_7_1_7"/>
( <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="c5">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="c5">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> <font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font> &amp; <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="c14">p9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c9">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="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="txt" href="#E6:42"><span class="lab"><font color="Green" title="E30">A14</font></span></a>, <a class="ref" href="lattice5.html#D6" target="_self" title="LATTICE5:def.6">Def6</a></span>;<br/><a NAME="E6:42_7_1_7"/><span class="kw">then </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="c14">p9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c5">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c9">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> <font color="Maroon" title="c14">p9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c9">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> <span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> &amp; <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="c14">p9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c9">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> <span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c14">p9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a>, <a class="ref" href="yellow_3.html#T3" title="YELLOW_3:th.3">YELLOW_3:3</a></span>;<br/><a NAME="E7:42_7_1_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A131</font></span>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="c14">p9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:42_7_1_7"><span class="lab"><font color="Green" title="E40">A130</font></span></a>, <a class="ref" href="orders_2.html#T26" title="ORDERS_2:th.26">ORDERS_2:26</a></span>;<br/><span class="lab"><font color="Green" title="E42">A132</font></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="c14">p9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p9</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></span>
<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="c14">p9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></span>
<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="c14">p9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_5.html#T1" title="YELLOW_5:th.1">YELLOW_5:1</a></span>
<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="c14">p9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></span>
<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="c14">p9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_5.html#T1" title="YELLOW_5:th.1">YELLOW_5:1</a></span>
<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="c14">p9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></span>
<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="c14">p9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></span>
;<br/><a NAME="E9:42_7_1_7"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="c14">p9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c10">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_7_1_7"><span class="lab"><font color="Green" title="E39">A129</font></span></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>;<br/><span class="kw">hence </span><a NAME="E10:42_7_1_7"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_7_1_7"><span class="lab"><font color="Green" title="E39">A129</font></span></a>, <a class="txt" href="#E7:42_7_1_7"><span class="lab"><font color="Green" title="E41">A131</font></span></a>, <a class="txt" href="#E8:42_7_1_7"><span class="lab"><font color="Green" title="E42">A132</font></span></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_7_1_8"><span class="kw">suppose </span></a><a NAME="E1:42_7_1_8"/><span class="lab"><font color="Green" title="E39">A133</font></span>: 
( <font color="Maroon" title="c11">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="c12">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"><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#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><a NAME="E2:42_7_1_8"/>
<font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p9</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">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="c14">p9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c5">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="ref" href="lattice5.html#D8" target="_self" title="LATTICE5:def.8">Def8</a></span>;<br/><a NAME="E3:42_7_1_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A134</font></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="c14">p9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</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> <font color="Maroon" title="c14">p9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c5">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_1.html#T3" title="WAYBEL_1:th.3">WAYBEL_1:3</a></span>;<br/><a NAME="E4:42_7_1_8"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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> <font color="Maroon" title="c14">p9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_7_1_8"><span class="lab"><font color="Green" title="E39">A133</font></span></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>;<br/><a NAME="E5:42_7_1_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A135</font></span>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="c14">p9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c5">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42_7_1_8"><span class="lab"><font color="Green" title="E40">A134</font></span></a>, <a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></span>;<br/><span class="lab"><font color="Green" title="E42">A136</font></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="c14">p9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p9</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></span>
<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="c14">p9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></span>
<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="c14">p9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_5.html#T1" title="YELLOW_5:th.1">YELLOW_5:1</a></span>
<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="c14">p9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></span>
;<br/><a NAME="E7:42_7_1_8"/><span class="lab"><font color="Green" title="E43">A137</font></span>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="c14">p9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c10">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_7_1_8"><span class="lab"><font color="Green" title="E39">A133</font></span></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>;<br/><a NAME="E8:42_7_1_8"/><span class="lab"><font color="Green" title="E44">A138</font></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="c14">p9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></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="c14">p9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
 ;<br/><a NAME="E9:42_7_1_8"/>
( <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="c5">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="c5">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> <font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font> &amp; <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="c14">p9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c5">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> <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> <font color="Maroon" title="c14">p9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></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="c5">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="txt" href="#E6:42"><span class="lab"><font color="Green" title="E30">A14</font></span></a>, <a class="ref" href="lattice5.html#D6" target="_self" title="LATTICE5:def.6">Def6</a>, <a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></span>;<br/><a NAME="E10:42_7_1_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A139</font></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="c14">p9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c5">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</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> <font color="Maroon" title="c14">p9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:42_7_1_8"><span class="lab"><font color="Green" title="E44">A138</font></span></a>, <a class="ref" href="yellow_3.html#T3" title="YELLOW_3:th.3">YELLOW_3:3</a></span>;<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="c14">p9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</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="c14">p9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c9">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="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_5.html#T1" title="YELLOW_5:th.1">YELLOW_5:1</a></span>
<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="c14">p9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font> <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="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></span>
<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="c14">p9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></span>
<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="c14">p9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></span>
<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="c14">p9</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_5.html#T1" title="YELLOW_5:th.1">YELLOW_5:1</a></span>
<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="c14">p9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></span>
;<br/><a NAME="E12:42_7_1_8"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="c14">p9</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">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="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:42_7_1_8"><span class="lab"><font color="Green" title="E45">A139</font></span></a>, <a class="txt" href="#E5:42_7_1_8"><span class="lab"><font color="Green" title="E41">A135</font></span></a>, <a class="ref" href="orders_2.html#T26" title="ORDERS_2:th.26">ORDERS_2:26</a></span>;<br/><span class="kw">hence </span><a NAME="E13:42_7_1_8"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_7_1_8"><span class="lab"><font color="Green" title="E39">A133</font></span></a>, <a class="txt" href="#E7:42_7_1_8"><span class="lab"><font color="Green" title="E43">A137</font></span></a>, <a class="txt" href="#E6:42_7_1_8"><span class="lab"><font color="Green" title="E42">A136</font></span></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_7_1_9"><span class="kw">suppose </span></a><a NAME="E1:42_7_1_9"/><span class="lab"><font color="Green" title="E39">A140</font></span>: 
( <font color="Maroon" title="c11">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="c12">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"><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#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">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"><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#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><span class="kw">then </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_1.html#T4" title="WAYBEL_1:th.4">WAYBEL_1:4</a></span>
;<br/><span class="kw">hence </span><a NAME="E3:42_7_1_9"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_7_1_9"><span class="lab"><font color="Green" title="E39">A140</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E13:42"/><span class="lab"><font color="Green" title="E37">A141</font></span>: 
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="lattice5.html#K6" title="LATTICE5:func.6">new_set</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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">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="42_8"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">p</font>, <font color="Maroon" title="c12">q</font>, <font color="Maroon" title="c13">u</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice5.html#K6" title="LATTICE5:func.6">new_set</a> <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">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="c12">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">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> implies <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:42_8"/>
( <font color="Maroon" title="c11">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="c12">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">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">p9</font> = <font color="Maroon" title="c11">p</font>, <font color="Maroon" title="c15">q9</font> = <font color="Maroon" title="c12">q</font>, <font color="Maroon" title="c16">u9</font> = <font color="Maroon" title="c13">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="E3:42_8"/><span class="lab"><font color="Green" title="E38">A142</font></span>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">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="c15">q9</font>,<font color="Maroon" title="c16">u9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>;<br/>
<a NAME="E4:42_8"/>
( <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="c14">p9</font>,<font color="Maroon" title="c16">u9</font> &amp; <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="c14">p9</font>,<font color="Maroon" title="c15">q9</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:42_8"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="txt" href="#E3:42_8"><span class="lab"><font color="Green" title="E38">A142</font></span></a>, <a class="ref" href="lattice5.html#D8" target="_self" title="LATTICE5:def.8">Def8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E14:42"/>
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="lattice5.html#K6" title="LATTICE5:func.6">new_set</a> <font color="Maroon" title="c1">A</font> holds  <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">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="42_9"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">p</font>, <font color="Maroon" title="c12">q</font>, <font color="Maroon" title="c13">u</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice5.html#K6" title="LATTICE5:func.6">new_set</a> <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_9_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:42_9_1"/>
( ( <font color="Maroon" title="c11">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="c12">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">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="c11">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="c12">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">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> ) or ( <font color="Maroon" title="c11">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="c12">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> &amp; <font color="Maroon" title="c13">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="c11">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="c12">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> ) or ( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> &amp; <font color="Maroon" title="c12">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">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="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> &amp; <font color="Maroon" title="c12">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">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> ) or ( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> &amp; <font color="Maroon" title="c13">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="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_9_1_1"><span class="kw">suppose </span></a><a NAME="E1:42_9_1_1"/>
( <font color="Maroon" title="c11">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="c12">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">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:42_9_1_1"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:42"><span class="lab"><font color="Green" title="E37">A141</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_9_1_2"><span class="kw">suppose </span></a><a NAME="E1:42_9_1_2"/>
( <font color="Maroon" title="c11">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="c12">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">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:42_9_1_2"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:42"><span class="lab"><font color="Green" title="E33">A54</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_9_1_3"><span class="kw">suppose </span></a><a NAME="E1:42_9_1_3"/>
( <font color="Maroon" title="c11">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="c12">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:42_9_1_3"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:42"><span class="lab"><font color="Green" title="E29">A3</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_9_1_4"><span class="kw">suppose </span></a><a NAME="E1:42_9_1_4"/>
( <font color="Maroon" title="c11">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="c12">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:42_9_1_4"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:42"><span class="lab"><font color="Green" title="E36">A114</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_9_1_5"><span class="kw">suppose </span></a><a NAME="E1:42_9_1_5"/>
( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> &amp; <font color="Maroon" title="c12">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">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:42_9_1_5"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:42"><span class="lab"><font color="Green" title="E32">A42</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_9_1_6"><span class="kw">suppose </span></a><a NAME="E1:42_9_1_6"/>
( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> &amp; <font color="Maroon" title="c12">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">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:42_9_1_6"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:42"><span class="lab"><font color="Green" title="E35">A98</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_9_1_7"><span class="kw">suppose </span></a><a NAME="E1:42_9_1_7"/>
( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:42_9_1_7"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:42"><span class="lab"><font color="Green" title="E31">A15</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_9_1_8"><span class="kw">suppose </span></a><a NAME="E1:42_9_1_8"/>
( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:42_9_1_8"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">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="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:42"><span class="lab"><font color="Green" title="E34">A65</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E15:42"/>
 <a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">q</font> is  <a href="lattice5.html#V3" title="LATTICE5:attr.3">u.t.i.</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattice5.html#D8" target="_self" title="LATTICE5:def.8">Def8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
