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<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="firebrick">::  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>  st <font color="Olive" title="b1">L</font> is  <a href="yellow11.html#V1" title="YELLOW11:attr.1">modular</a>  holds <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#K3" title="LATTICE5:func.3">.</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="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">L</font> is  <a href="yellow11.html#V1" title="YELLOW11:attr.1">modular</a>  implies  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#K3" title="LATTICE5:func.3">.</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="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</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">assume </span><a NAME="E1:21"/><span class="lab"><font color="Green" title="E16">A1</font></span>: 
<font color="Maroon" title="c2">L</font> is  <a href="yellow11.html#V1" title="YELLOW11:attr.1">modular</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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#K3" title="LATTICE5:func.3">.</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="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</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">reconsider </span><font color="Maroon" title="c3">B</font> = <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<span class="kw">let </span><font color="Maroon" title="c4">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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">d</font> is  <a href="lattice5.html#V1" title="LATTICE5:attr.1">symmetric</a>  &amp; <font color="Maroon" title="c4">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="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</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="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">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="E3:21"/><span class="lab"><font color="Green" title="E17">A2</font></span>: 
<font color="Maroon" title="c4">d</font> is  <a href="lattice5.html#V1" title="LATTICE5:attr.1">symmetric</a> 
 <span class="kw">and </span><br/><a NAME="E4:21"/><span class="lab"><font color="Green" title="E18">A3</font></span>: 
<font color="Maroon" title="c4">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="firebrick">::  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="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</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="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">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">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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</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="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">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">a</font> = <font color="Maroon" title="c5">q</font> <a href="mcart_1.html#K10" title="MCART_1:func.10">`3</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">b</font> = <font color="Maroon" title="c5">q</font> <a href="mcart_1.html#K11" title="MCART_1:func.11">`4</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">f</font> =  <a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>);<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">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="c12">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="E6:21"/><span class="lab"><font color="Green" title="E19">A4</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="lattice8.html#K1" title="LATTICE8:func.1">new_set2</a> <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Olive" title="b2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Olive" title="b3">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> holds <br/><span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Olive" title="b1">p</font>,<font color="Olive" title="b3">u</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</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="21_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">p</font>, <font color="Maroon" title="c14">q</font>, <font color="Maroon" title="c15">u</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice8.html#K1" title="LATTICE8:func.1">new_set2</a> <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> implies <span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:21_1"/><span class="lab"><font color="Green" title="E20">A5</font></span>: 
( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_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="E1:21_1_1"/>
( ( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_1"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:21_1_1_1"/><span class="lab"><font color="Green" title="E21">A6</font></span>: 
( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span></span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c16">p9</font> = <font color="Maroon" title="c13">p</font>, <font color="Maroon" title="c17">q9</font> = <font color="Maroon" title="c14">q</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> ;<br/><a NAME="E3:21_1_1_1"/><span class="lab"><font color="Green" title="E22">A7</font></span>: 
<span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">p9</font>,<font color="Maroon" title="c17">q9</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattice8.html#D4" target="_self" title="LATTICE8:def.4">Def5</a></span>;<br/><a NAME="E4:21_1_1_1"/>
<font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">p9</font>,<font color="Maroon" title="c17">q9</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c17">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="#E4:21"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="ref" href="lattice5.html#D7" title="LATTICE5:def.7">LATTICE5:def 7</a></span>;<br/><a NAME="E5:21_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A8</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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="c11">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">p9</font>,<font color="Maroon" title="c17">q9</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c17">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="c11">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_1.html#T2" title="WAYBEL_1:th.2">WAYBEL_1:2</a></span>;<br/><a NAME="E6:21_1_1_1"/>
( <span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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="c11">a</font> &amp; <span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c17">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="c11">a</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_1_1_1"><span class="lab"><font color="Green" title="E21">A6</font></span></a>, <a class="ref" href="lattice8.html#D4" target="_self" title="LATTICE8:def.4">Def5</a></span>;<br/><span class="kw">hence </span><a NAME="E7:21_1_1_1"/>
<span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21_1_1_1"><span class="lab"><font color="Green" title="E22">A7</font></span></a>, <a class="txt" href="#E5:21_1_1_1"><span class="lab"><font color="Green" title="E23">A8</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="firebrick">::  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="21_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:21_1_1_2"/><span class="lab"><font color="Green" title="E21">A9</font></span>: 
( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span></span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c16">p9</font> = <font color="Maroon" title="c13">p</font>, <font color="Maroon" title="c17">q9</font> = <font color="Maroon" title="c14">q</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> ;<br/><a NAME="E3:21_1_1_2"/><span class="lab"><font color="Green" title="E22">A10</font></span>: 
<span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">p9</font>,<font color="Maroon" title="c17">q9</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattice8.html#D4" target="_self" title="LATTICE8:def.4">Def5</a></span>;<br/><a NAME="E4:21_1_1_2"/>
<font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">p9</font>,<font color="Maroon" title="c17">q9</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c17">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="#E4:21"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="ref" href="lattice5.html#D7" title="LATTICE5:def.7">LATTICE5:def 7</a></span>;<br/><a NAME="E5:21_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A11</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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="c11">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">p9</font>,<font color="Maroon" title="c17">q9</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c17">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="c11">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_1.html#T2" title="WAYBEL_1:th.2">WAYBEL_1:2</a></span>;<br/><a NAME="E6:21_1_1_2"/>
( <span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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="c11">a</font> &amp; <span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c17">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="c11">a</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_1_1_2"><span class="lab"><font color="Green" title="E21">A9</font></span></a>, <a class="ref" href="lattice8.html#D4" target="_self" title="LATTICE8:def.4">Def5</a></span>;<br/><span class="kw">hence </span><a NAME="E7:21_1_1_2"/>
<span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21_1_1_2"><span class="lab"><font color="Green" title="E22">A10</font></span></a>, <a class="txt" href="#E5:21_1_1_2"><span class="lab"><font color="Green" title="E23">A11</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="firebrick">::  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="E7:21"/><span class="lab"><font color="Green" title="E20">A12</font></span>: 
<font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</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="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">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="E8:21"/><span class="lab"><font color="Green" title="E21">A13</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="lattice8.html#K1" title="LATTICE8:func.1">new_set2</a> <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Olive" title="b2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">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="c3">B</font> holds <br/><span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Olive" title="b1">p</font>,<font color="Olive" title="b3">u</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</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="21_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">p</font>, <font color="Maroon" title="c14">q</font>, <font color="Maroon" title="c15">u</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice8.html#K1" title="LATTICE8:func.1">new_set2</a> <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> implies <span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:21_2"/><span class="lab"><font color="Green" title="E22">A14</font></span>: 
( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_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="E1:21_2_1"/>
( ( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_2"><span class="lab"><font color="Green" title="E22">A14</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:21_2_1_1"/><span class="lab"><font color="Green" title="E23">A15</font></span>: 
( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">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="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span> = 
<span class="p1">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice8.html#D4" target="_self" title="LATTICE8:def.4">Def5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</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="kw">hence </span><a NAME="E3:21_2_1_1"/>
<span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_2_1_1"><span class="lab"><font color="Green" title="E23">A15</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="21_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:21_2_1_2"/><span class="lab"><font color="Green" title="E23">A16</font></span>: 
( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span></span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c16">p9</font> = <font color="Maroon" title="c13">p</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> ;<br/><a NAME="E3:21_2_1_2"/>
<font color="Maroon" title="c11">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c11">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</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="ref" href="yellow_0.html#T22" title="YELLOW_0:th.22">YELLOW_0:22</a></span>;<br/><a NAME="E4:21_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A17</font></span>: 
<font color="Maroon" title="c11">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<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 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="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="ref" href="yellow11.html#D3" title="YELLOW11:def.3">YELLOW11:def 3</a></span>;<br/><a NAME="E5:21_2_1_2"/>
<font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</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="#E4:21"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="ref" href="lattice5.html#D7" title="LATTICE5:def.7">LATTICE5:def 7</a></span>;<br/><a NAME="E6:21_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A18</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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="c11">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</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="c11">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_5.html#T7" title="YELLOW_5:th.7">YELLOW_5:7</a></span>;<br/><a NAME="E7:21_2_1_2"/>
<font color="Maroon" title="c11">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c11">a</font>
 ;<br/><a NAME="E8:21_2_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</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="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:21"><span class="lab"><font color="Green" title="E20">A12</font></span></a>, <a class="ref" href="yellow_3.html#T3" title="YELLOW_3:th.3">YELLOW_3:3</a></span>;<br/><a NAME="E9:21_2_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</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> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</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> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</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> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_5.html#T6" title="YELLOW_5:th.6">YELLOW_5:6</a></span>;<br/><a NAME="E10:21_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A19</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</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="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</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> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</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> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<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 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="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<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 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="c11">a</font></span>)</span></span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<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 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="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_1.html#T2" title="WAYBEL_1:th.2">WAYBEL_1:2</a>, <a class="ref" href="yellow_5.html#T2" title="YELLOW_5:th.2">YELLOW_5:2</a></span>;<br/><a NAME="E11:21_2_1_2"/>
( <span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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="c11">a</font> &amp; <span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</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> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c12">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_2_1_2"><span class="lab"><font color="Green" title="E23">A16</font></span></a>, <a class="ref" href="lattice8.html#D4" target="_self" title="LATTICE8:def.4">Def5</a></span>;<br/><span class="kw">then </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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="c11">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<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 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></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:21_2_1_2"><span class="lab"><font color="Green" title="E24">A17</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"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<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 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="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c12">b</font></span>)</span></span>)</span>
<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="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<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 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="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span></span>)</span>
<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="E13:21_2_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</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="c11">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:21_2_1_2"><span class="lab"><font color="Green" title="E26">A19</font></span></a>, <a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></span>;<br/><a NAME="E14:21_2_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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="c11">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:21_2_1_2"><span class="lab"><font color="Green" title="E25">A18</font></span></a>, <a class="ref" href="orders_2.html#T3" title="ORDERS_2:th.3">ORDERS_2:3</a></span>;<br/><span class="kw">hence </span><a NAME="E15:21_2_1_2"/>
<span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_2_1_2"><span class="lab"><font color="Green" title="E23">A16</font></span></a>, <a class="ref" href="lattice8.html#D4" target="_self" title="LATTICE8:def.4">Def5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="21_2_1_3"><span class="kw">suppose </span></a><a NAME="E1:21_2_1_3"/><span class="lab"><font color="Green" title="E23">A20</font></span>: 
( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span></span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c16">p9</font> = <font color="Maroon" title="c13">p</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> ;<br/><a NAME="E3:21_2_1_3"/>
<font color="Maroon" title="c11">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c11">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</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="ref" href="yellow_0.html#T22" title="YELLOW_0:th.22">YELLOW_0:22</a></span>;<br/><a NAME="E4:21_2_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A21</font></span>: 
<font color="Maroon" title="c11">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<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 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="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="ref" href="yellow11.html#D3" title="YELLOW11:def.3">YELLOW11:def 3</a></span>;<br/><a NAME="E5:21_2_1_3"/>
<font color="Maroon" title="c11">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c11">a</font>
 ;<br/><a NAME="E6:21_2_1_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</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="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:21"><span class="lab"><font color="Green" title="E20">A12</font></span></a>, <a class="ref" href="yellow_3.html#T3" title="YELLOW_3:th.3">YELLOW_3:3</a></span>;<br/><a NAME="E7:21_2_1_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</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> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</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> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</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> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_5.html#T6" title="YELLOW_5:th.6">YELLOW_5:6</a></span>;<br/><a NAME="E8:21_2_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A22</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</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="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</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> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</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> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<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 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="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<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 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="c11">a</font></span>)</span></span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<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 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="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_1.html#T2" title="WAYBEL_1:th.2">WAYBEL_1:2</a>, <a class="ref" href="yellow_5.html#T2" title="YELLOW_5:th.2">YELLOW_5:2</a></span>;<br/><a NAME="E9:21_2_1_3"/>
<font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</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>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="ref" href="lattice5.html#D5" title="LATTICE5:def.5">LATTICE5:def 5</a></span>;<br/><a NAME="E10:21_2_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</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="#E4:21"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="ref" href="lattice5.html#D7" title="LATTICE5:def.7">LATTICE5:def 7</a></span>;<br/><a NAME="E11:21_2_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A23</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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="c11">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</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="c11">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_5.html#T7" title="YELLOW_5:th.7">YELLOW_5:7</a></span>;<br/><a NAME="E12:21_2_1_3"/>
( <span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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="c11">a</font> &amp; <span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</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> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c12">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_2_1_3"><span class="lab"><font color="Green" title="E23">A20</font></span></a>, <a class="ref" href="lattice8.html#D4" target="_self" title="LATTICE8:def.4">Def5</a></span>;<br/><span class="kw">then </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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="c11">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<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 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></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:21_2_1_3"><span class="lab"><font color="Green" title="E24">A21</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"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<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 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="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c12">b</font></span>)</span></span>)</span>
<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="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<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 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="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span></span>)</span>
<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="E14:21_2_1_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</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="c11">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:21_2_1_3"><span class="lab"><font color="Green" title="E25">A22</font></span></a>, <a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></span>;<br/><a NAME="E15:21_2_1_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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="c11">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:21_2_1_3"><span class="lab"><font color="Green" title="E26">A23</font></span></a>, <a class="ref" href="orders_2.html#T3" title="ORDERS_2:th.3">ORDERS_2:3</a></span>;<br/><span class="kw">hence </span><a NAME="E16:21_2_1_3"/>
<span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_2_1_3"><span class="lab"><font color="Green" title="E23">A20</font></span></a>, <a class="ref" href="lattice8.html#D4" target="_self" title="LATTICE8:def.4">Def5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="21_2_1_4"><span class="kw">suppose </span></a><a NAME="E1:21_2_1_4"/><span class="lab"><font color="Green" title="E23">A24</font></span>: 
( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">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="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span> = 
<span class="p1">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice8.html#D4" target="_self" title="LATTICE8:def.4">Def5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</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="kw">hence </span><a NAME="E3:21_2_1_4"/>
<span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_2_1_4"><span class="lab"><font color="Green" title="E23">A24</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="E9:21"/><span class="lab"><font color="Green" title="E22">A25</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="lattice8.html#K1" title="LATTICE8:func.1">new_set2</a> <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">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="c3">B</font> holds <br/><span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Olive" title="b1">p</font>,<font color="Olive" title="b3">u</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</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="21_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">p</font>, <font color="Maroon" title="c14">q</font>, <font color="Maroon" title="c15">u</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice8.html#K1" title="LATTICE8:func.1">new_set2</a> <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> implies <span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:21_3"/><span class="lab"><font color="Green" title="E23">A26</font></span>: 
( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_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="E1:21_3_1"/>
( ( <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_3"><span class="lab"><font color="Green" title="E23">A26</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:21_3_1_1"/>
( <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span></span><br/><div class="add"><a NAME="E2:21_3_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c2">L</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattice8.html#D4" target="_self" title="LATTICE8:def.4">Def5</a></span>;<br/><span class="kw">hence </span><a NAME="E3:21_3_1_1"/>
<span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">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="firebrick">::  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="21_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:21_3_1_2"/><span class="lab"><font color="Green" title="E24">A27</font></span>: 
( <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span></span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c16">q9</font> = <font color="Maroon" title="c14">q</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> ;<br/><a NAME="E3:21_3_1_2"/>
<font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">q9</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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</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>,<font color="Maroon" title="c16">q9</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="ref" href="lattice5.html#D5" title="LATTICE5:def.5">LATTICE5:def 5</a></span>;<br/><a NAME="E4:21_3_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A28</font></span>: 
<font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</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="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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="#E4:21"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="ref" href="lattice5.html#D7" title="LATTICE5:def.7">LATTICE5:def 7</a></span>;<br/><a NAME="E5:21_3_1_2"/>
<span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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="c11">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_3_1_2"><span class="lab"><font color="Green" title="E24">A27</font></span></a>, <a class="ref" href="lattice8.html#D4" target="_self" title="LATTICE8:def.4">Def5</a></span>;<br/><span class="kw">then </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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="c11">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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="c11">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_3_1_2"><span class="lab"><font color="Green" title="E24">A27</font></span></a>, <a class="ref" href="lattice8.html#D4" target="_self" title="LATTICE8:def.4">Def5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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"><span class="p3">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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="c11">a</font></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">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="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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> <span class="p3">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font>
<span 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="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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="c11">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">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="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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="c11">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/><a NAME="E7:21_3_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A29</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</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="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:21_3_1_2"><span class="lab"><font color="Green" title="E25">A28</font></span></a>, <a class="ref" href="yellow_5.html#T7" title="YELLOW_5:th.7">YELLOW_5:7</a></span>;<br/><a NAME="E8:21_3_1_2"/><span class="lab"><font color="Green" title="E27">A30</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</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> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c12">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</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="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T23" title="YELLOW_0:th.23">YELLOW_0:23</a></span>;<br/><a NAME="E9:21_3_1_2"/>
<span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</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> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c12">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_3_1_2"><span class="lab"><font color="Green" title="E24">A27</font></span></a>, <a class="ref" href="lattice8.html#D4" target="_self" title="LATTICE8:def.4">Def5</a></span>;<br/><span class="kw">hence </span><a NAME="E10:21_3_1_2"/>
<span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:21_3_1_2"><span class="lab"><font color="Green" title="E26">A29</font></span></a>, <a class="txt" href="#E8:21_3_1_2"><span class="lab"><font color="Green" title="E27">A30</font></span></a>, <a class="ref" href="orders_2.html#T3" title="ORDERS_2:th.3">ORDERS_2:3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="21_3_1_3"><span class="kw">suppose </span></a><a NAME="E1:21_3_1_3"/><span class="lab"><font color="Green" title="E24">A31</font></span>: 
( <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span></span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c16">q9</font> = <font color="Maroon" title="c14">q</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> ;<br/><a NAME="E3:21_3_1_3"/>
<font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">q9</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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</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>,<font color="Maroon" title="c16">q9</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="ref" href="lattice5.html#D5" title="LATTICE5:def.5">LATTICE5:def 5</a></span>;<br/><a NAME="E4:21_3_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A32</font></span>: 
<font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</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="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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="#E4:21"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="ref" href="lattice5.html#D7" title="LATTICE5:def.7">LATTICE5:def 7</a></span>;<br/><a NAME="E5:21_3_1_3"/>
<span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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="c11">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_3_1_3"><span class="lab"><font color="Green" title="E24">A31</font></span></a>, <a class="ref" href="lattice8.html#D4" target="_self" title="LATTICE8:def.4">Def5</a></span>;<br/><span class="kw">then </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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="c11">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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="c11">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_3_1_3"><span class="lab"><font color="Green" title="E24">A31</font></span></a>, <a class="ref" href="lattice8.html#D4" target="_self" title="LATTICE8:def.4">Def5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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"><span class="p3">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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="c11">a</font></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">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="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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> <span class="p3">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font>
<span 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="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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="c11">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">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="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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="c11">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/><a NAME="E7:21_3_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A33</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</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="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:21_3_1_3"><span class="lab"><font color="Green" title="E25">A32</font></span></a>, <a class="ref" href="yellow_5.html#T7" title="YELLOW_5:th.7">YELLOW_5:7</a></span>;<br/><a NAME="E8:21_3_1_3"/><span class="lab"><font color="Green" title="E27">A34</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</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> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c12">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</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="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T23" title="YELLOW_0:th.23">YELLOW_0:23</a></span>;<br/><a NAME="E9:21_3_1_3"/>
<span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</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> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c12">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_3_1_3"><span class="lab"><font color="Green" title="E24">A31</font></span></a>, <a class="ref" href="lattice8.html#D4" target="_self" title="LATTICE8:def.4">Def5</a></span>;<br/><span class="kw">hence </span><a NAME="E10:21_3_1_3"/>
<span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:21_3_1_3"><span class="lab"><font color="Green" title="E26">A33</font></span></a>, <a class="txt" href="#E8:21_3_1_3"><span class="lab"><font color="Green" title="E27">A34</font></span></a>, <a class="ref" href="orders_2.html#T3" title="ORDERS_2:th.3">ORDERS_2:3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="21_3_1_4"><span class="kw">suppose </span></a><a NAME="E1:21_3_1_4"/>
( <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span></span><br/><div class="add"><a NAME="E2:21_3_1_4"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c2">L</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattice8.html#D4" target="_self" title="LATTICE8:def.4">Def5</a></span>;<br/><span class="kw">hence </span><a NAME="E3:21_3_1_4"/>
<span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">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="firebrick">::  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="E10:21"/><span class="lab"><font color="Green" title="E23">A35</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="lattice8.html#K1" title="LATTICE8:func.1">new_set2</a> <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">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="c3">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="c3">B</font> holds <br/><span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Olive" title="b1">p</font>,<font color="Olive" title="b3">u</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</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="21_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">p</font>, <font color="Maroon" title="c14">q</font>, <font color="Maroon" title="c15">u</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice8.html#K1" title="LATTICE8:func.1">new_set2</a> <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> implies <span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:21_4"/><span class="lab"><font color="Green" title="E24">A36</font></span>: 
( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_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="E1:21_4_1"/>
( ( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_4"><span class="lab"><font color="Green" title="E24">A36</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_4_1_1"><span class="kw">suppose </span></a><a NAME="E1:21_4_1_1"/><span class="lab"><font color="Green" title="E25">A37</font></span>: 
( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span></span><br/><div class="add"><a NAME="E2:21_4_1_1"/>
 <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c2">L</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">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="E3:21_4_1_1"/>
<span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_4_1_1"><span class="lab"><font color="Green" title="E25">A37</font></span></a>, <a class="ref" href="lattice8.html#D4" target="_self" title="LATTICE8:def.4">Def5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="21_4_1_2"><span class="kw">suppose </span></a><a NAME="E1:21_4_1_2"/><span class="lab"><font color="Green" title="E25">A38</font></span>: 
( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">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="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span> = 
<span class="p1">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice8.html#D4" target="_self" title="LATTICE8:def.4">Def5</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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<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 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="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c12">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_4_1_2"><span class="lab"><font color="Green" title="E25">A38</font></span></a>, <a class="ref" href="lattice8.html#D4" target="_self" title="LATTICE8:def.4">Def5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</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> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c12">b</font>
<span 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="kw">hence </span><a NAME="E3:21_4_1_2"/>
<span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_4_1_2"><span class="lab"><font color="Green" title="E25">A38</font></span></a>, <a class="ref" href="lattice8.html#D4" target="_self" title="LATTICE8:def.4">Def5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="21_4_1_3"><span class="kw">suppose </span></a><a NAME="E1:21_4_1_3"/><span class="lab"><font color="Green" title="E25">A39</font></span>: 
( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span></span><br/><div class="add"><a NAME="E2:21_4_1_3"/>
 <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c2">L</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">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="E3:21_4_1_3"/>
<span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_4_1_3"><span class="lab"><font color="Green" title="E25">A39</font></span></a>, <a class="ref" href="lattice8.html#D4" target="_self" title="LATTICE8:def.4">Def5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="21_4_1_4"><span class="kw">suppose </span></a><a NAME="E1:21_4_1_4"/><span class="lab"><font color="Green" title="E25">A40</font></span>: 
( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">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="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<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 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="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice8.html#D4" target="_self" title="LATTICE8:def.4">Def5</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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<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 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="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c12">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_4_1_4"><span class="lab"><font color="Green" title="E25">A40</font></span></a>, <a class="ref" href="lattice8.html#D4" target="_self" title="LATTICE8:def.4">Def5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</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> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c12">b</font>
<span 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="kw">hence </span><a NAME="E3:21_4_1_4"/>
<span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_4_1_4"><span class="lab"><font color="Green" title="E25">A40</font></span></a>, <a class="ref" href="lattice8.html#D4" target="_self" title="LATTICE8:def.4">Def5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="21_4_1_5"><span class="kw">suppose </span></a><a NAME="E1:21_4_1_5"/><span class="lab"><font color="Green" title="E25">A41</font></span>: 
( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">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="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<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 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="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice8.html#D4" target="_self" title="LATTICE8:def.4">Def5</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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<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 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="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c12">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_4_1_5"><span class="lab"><font color="Green" title="E25">A41</font></span></a>, <a class="ref" href="lattice8.html#D4" target="_self" title="LATTICE8:def.4">Def5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</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> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c12">b</font>
<span 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="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_4_1_5"><span class="lab"><font color="Green" title="E25">A41</font></span></a>, <a class="ref" href="lattice8.html#D4" target="_self" title="LATTICE8:def.4">Def5</a></span>
;<br/><span class="kw">hence </span><a NAME="E3:21_4_1_5"/>
<span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_4_1_5"><span class="lab"><font color="Green" title="E25">A41</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="21_4_1_6"><span class="kw">suppose </span></a><a NAME="E1:21_4_1_6"/><span class="lab"><font color="Green" title="E25">A42</font></span>: 
( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span></span><br/><div class="add"><a NAME="E2:21_4_1_6"/>
 <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c2">L</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">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="E3:21_4_1_6"/>
<span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_4_1_6"><span class="lab"><font color="Green" title="E25">A42</font></span></a>, <a class="ref" href="lattice8.html#D4" target="_self" title="LATTICE8:def.4">Def5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="21_4_1_7"><span class="kw">suppose </span></a><a NAME="E1:21_4_1_7"/><span class="lab"><font color="Green" title="E25">A43</font></span>: 
( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">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="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span> = 
<span class="p1">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice8.html#D4" target="_self" title="LATTICE8:def.4">Def5</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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<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 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="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c12">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_4_1_7"><span class="lab"><font color="Green" title="E25">A43</font></span></a>, <a class="ref" href="lattice8.html#D4" target="_self" title="LATTICE8:def.4">Def5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</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> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c12">b</font>
<span 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="kw">hence </span><a NAME="E3:21_4_1_7"/>
<span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_4_1_7"><span class="lab"><font color="Green" title="E25">A43</font></span></a>, <a class="ref" href="lattice8.html#D4" target="_self" title="LATTICE8:def.4">Def5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="21_4_1_8"><span class="kw">suppose </span></a><a NAME="E1:21_4_1_8"/><span class="lab"><font color="Green" title="E25">A44</font></span>: 
( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span></span><br/><div class="add"><a NAME="E2:21_4_1_8"/>
 <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c2">L</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">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="E3:21_4_1_8"/>
<span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_4_1_8"><span class="lab"><font color="Green" title="E25">A44</font></span></a>, <a class="ref" href="lattice8.html#D4" target="_self" title="LATTICE8:def.4">Def5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="E11:21"/><span class="lab"><font color="Green" title="E24">A45</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="lattice8.html#K1" title="LATTICE8:func.1">new_set2</a> <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">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="c3">B</font> &amp; <font color="Olive" title="b3">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> holds <br/><span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Olive" title="b1">p</font>,<font color="Olive" title="b3">u</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</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="21_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">p</font>, <font color="Maroon" title="c14">q</font>, <font color="Maroon" title="c15">u</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice8.html#K1" title="LATTICE8:func.1">new_set2</a> <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> implies <span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:21_5"/><span class="lab"><font color="Green" title="E25">A46</font></span>: 
( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_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="E1:21_5_1"/>
( ( <font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_5"><span class="lab"><font color="Green" title="E25">A46</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_5_1_1"><span class="kw">suppose </span></a><a NAME="E1:21_5_1_1"/><span class="lab"><font color="Green" title="E26">A47</font></span>: 
( <font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">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="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span> = 
<span class="p1">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice8.html#D4" target="_self" title="LATTICE8:def.4">Def5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_5_1_1"><span class="lab"><font color="Green" title="E26">A47</font></span></a>, <a class="ref" href="waybel_1.html#T3" title="WAYBEL_1:th.3">WAYBEL_1:3</a></span>
;<br/><span class="kw">hence </span><a NAME="E3:21_5_1_1"/>
<span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="21_5_1_2"><span class="kw">suppose </span></a><a NAME="E1:21_5_1_2"/><span class="lab"><font color="Green" title="E26">A48</font></span>: 
( <font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span></span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c16">u9</font> = <font color="Maroon" title="c15">u</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> ;<br/><a NAME="E3:21_5_1_2"/>
<font color="Maroon" title="c11">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c11">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</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="ref" href="yellow_0.html#T22" title="YELLOW_0:th.22">YELLOW_0:22</a></span>;<br/><a NAME="E4:21_5_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A49</font></span>: 
<font color="Maroon" title="c11">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<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 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="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="ref" href="yellow11.html#D3" title="YELLOW11:def.3">YELLOW11:def 3</a></span>;<br/><a NAME="E5:21_5_1_2"/>
<font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</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="#E4:21"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="ref" href="lattice5.html#D7" title="LATTICE5:def.7">LATTICE5:def 7</a></span>;<br/><a NAME="E6:21_5_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A50</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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="c11">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</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="c11">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_5.html#T7" title="YELLOW_5:th.7">YELLOW_5:7</a></span>;<br/><a NAME="E7:21_5_1_2"/>
<font color="Maroon" title="c11">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c11">a</font>
 ;<br/><a NAME="E8:21_5_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</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="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:21"><span class="lab"><font color="Green" title="E20">A12</font></span></a>, <a class="ref" href="yellow_3.html#T3" title="YELLOW_3:th.3">YELLOW_3:3</a></span>;<br/><a NAME="E9:21_5_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</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> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</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> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</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> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_5.html#T6" title="YELLOW_5:th.6">YELLOW_5:6</a></span>;<br/><a NAME="E10:21_5_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A51</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</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="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</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> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</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> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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"><span class="p3">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<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 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="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<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 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="c11">a</font></span>)</span></span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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"><span class="p3">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<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 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="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_1.html#T2" title="WAYBEL_1:th.2">WAYBEL_1:2</a>, <a class="ref" href="yellow_5.html#T2" title="YELLOW_5:th.2">YELLOW_5:2</a></span>;<br/><a NAME="E11:21_5_1_2"/>
( <span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</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> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c12">b</font> &amp; <span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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="c11">a</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_5_1_2"><span class="lab"><font color="Green" title="E26">A48</font></span></a>, <a class="ref" href="lattice8.html#D4" target="_self" title="LATTICE8:def.4">Def5</a></span>;<br/><span class="kw">then </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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="c11">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<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 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></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:21_5_1_2"><span class="lab"><font color="Green" title="E27">A49</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"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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"><span class="p3">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<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 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="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c12">b</font></span>)</span></span>)</span>
<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="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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"><span class="p3">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<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 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="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span></span>)</span>
<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="E13:21_5_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</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="c11">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:21_5_1_2"><span class="lab"><font color="Green" title="E29">A51</font></span></a>, <a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></span>;<br/><a NAME="E14:21_5_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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="c11">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:21_5_1_2"><span class="lab"><font color="Green" title="E28">A50</font></span></a>, <a class="ref" href="orders_2.html#T3" title="ORDERS_2:th.3">ORDERS_2:3</a></span>;<br/><span class="kw">hence </span><a NAME="E15:21_5_1_2"/>
<span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_5_1_2"><span class="lab"><font color="Green" title="E26">A48</font></span></a>, <a class="ref" href="lattice8.html#D4" target="_self" title="LATTICE8:def.4">Def5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="21_5_1_3"><span class="kw">suppose </span></a><a NAME="E1:21_5_1_3"/><span class="lab"><font color="Green" title="E26">A52</font></span>: 
( <font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span></span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c16">u9</font> = <font color="Maroon" title="c15">u</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> ;<br/><a NAME="E3:21_5_1_3"/>
<font color="Maroon" title="c11">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c11">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</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="ref" href="yellow_0.html#T22" title="YELLOW_0:th.22">YELLOW_0:22</a></span>;<br/><a NAME="E4:21_5_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A53</font></span>: 
<font color="Maroon" title="c11">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<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 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="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="ref" href="yellow11.html#D3" title="YELLOW11:def.3">YELLOW11:def 3</a></span>;<br/><a NAME="E5:21_5_1_3"/>
<font color="Maroon" title="c11">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c11">a</font>
 ;<br/><a NAME="E6:21_5_1_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</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="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:21"><span class="lab"><font color="Green" title="E20">A12</font></span></a>, <a class="ref" href="yellow_3.html#T3" title="YELLOW_3:th.3">YELLOW_3:3</a></span>;<br/><a NAME="E7:21_5_1_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</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> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</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> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</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> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_5.html#T6" title="YELLOW_5:th.6">YELLOW_5:6</a></span>;<br/><a NAME="E8:21_5_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A54</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</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="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</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> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</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> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<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 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="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<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 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="c11">a</font></span>)</span></span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<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 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="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_1.html#T2" title="WAYBEL_1:th.2">WAYBEL_1:2</a>, <a class="ref" href="yellow_5.html#T2" title="YELLOW_5:th.2">YELLOW_5:2</a></span>;<br/><a NAME="E9:21_5_1_3"/>
<font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</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>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="ref" href="lattice5.html#D5" title="LATTICE5:def.5">LATTICE5:def 5</a></span>;<br/><a NAME="E10:21_5_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</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="#E4:21"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="ref" href="lattice5.html#D7" title="LATTICE5:def.7">LATTICE5:def 7</a></span>;<br/><a NAME="E11:21_5_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A55</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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="c11">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</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="c11">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_5.html#T7" title="YELLOW_5:th.7">YELLOW_5:7</a></span>;<br/><a NAME="E12:21_5_1_3"/>
( <span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</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> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c12">b</font> &amp; <span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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="c11">a</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_5_1_3"><span class="lab"><font color="Green" title="E26">A52</font></span></a>, <a class="ref" href="lattice8.html#D4" target="_self" title="LATTICE8:def.4">Def5</a></span>;<br/><span class="kw">then </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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"><span class="p2">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<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 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></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:21_5_1_3"><span class="lab"><font color="Green" title="E27">A53</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"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<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 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="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c12">b</font></span>)</span></span>)</span>
<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="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<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 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="c11">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font></span>)</span></span>)</span>
<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="E14:21_5_1_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</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="c11">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:21_5_1_3"><span class="lab"><font color="Green" title="E28">A54</font></span></a>, <a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></span>;<br/><a NAME="E15:21_5_1_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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="c11">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:21_5_1_3"><span class="lab"><font color="Green" title="E29">A55</font></span></a>, <a class="ref" href="orders_2.html#T3" title="ORDERS_2:th.3">ORDERS_2:3</a></span>;<br/><span class="kw">hence </span><a NAME="E16:21_5_1_3"/>
<span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_5_1_3"><span class="lab"><font color="Green" title="E26">A52</font></span></a>, <a class="ref" href="lattice8.html#D4" target="_self" title="LATTICE8:def.4">Def5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="21_5_1_4"><span class="kw">suppose </span></a><a NAME="E1:21_5_1_4"/><span class="lab"><font color="Green" title="E26">A56</font></span>: 
( <font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">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="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span> = 
<span class="p1">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice8.html#D4" target="_self" title="LATTICE8:def.4">Def5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_5_1_4"><span class="lab"><font color="Green" title="E26">A56</font></span></a>, <a class="ref" href="waybel_1.html#T3" title="WAYBEL_1:th.3">WAYBEL_1:3</a></span>
;<br/><span class="kw">hence </span><a NAME="E3:21_5_1_4"/>
<span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="E12:21"/><span class="lab"><font color="Green" title="E25">A57</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="lattice8.html#K1" title="LATTICE8:func.1">new_set2</a> <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Olive" title="b2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Olive" title="b3">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> holds <br/><span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Olive" title="b1">p</font>,<font color="Olive" title="b3">u</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</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="21_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">p</font>, <font color="Maroon" title="c14">q</font>, <font color="Maroon" title="c15">u</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice8.html#K1" title="LATTICE8:func.1">new_set2</a> <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> implies <span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:21_6"/><span class="lab"><font color="Green" title="E26">A58</font></span>: 
<font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font>
 <span class="kw">and </span><br/><a NAME="E2:21_6"/><span class="lab"><font color="Green" title="E27">A59</font></span>: 
( <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c16">q9</font> = <font color="Maroon" title="c14">q</font>, <font color="Maroon" title="c17">u9</font> = <font color="Maroon" title="c15">u</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21_6"><span class="lab"><font color="Green" title="E27">A59</font></span></a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_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="E1:21_6_1"/>
( ( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> ) or ( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_6"><span class="lab"><font color="Green" title="E26">A58</font></span></a>, <a class="txt" href="#E2:21_6"><span class="lab"><font color="Green" title="E27">A59</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_6_1_1"><span class="kw">suppose </span></a><a NAME="E1:21_6_1_1"/><span class="lab"><font color="Green" title="E28">A60</font></span>: 
( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span></span><br/><div class="add"><a NAME="E2:21_6_1_1"/>
<font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c17">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="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c17">u9</font>,<font color="Maroon" title="c16">q9</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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="#E4:21"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="ref" href="lattice5.html#D7" title="LATTICE5:def.7">LATTICE5:def 7</a></span>;<br/><a NAME="E3:21_6_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c17">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="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">q9</font>,<font color="Maroon" title="c17">u9</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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="#E3:21"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="ref" href="lattice5.html#D5" title="LATTICE5:def.5">LATTICE5:def 5</a></span>;<br/><a NAME="E4:21_6_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c17">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="c11">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">q9</font>,<font color="Maroon" title="c17">u9</font>)</span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_1.html#T2" title="WAYBEL_1:th.2">WAYBEL_1:2</a></span>;<br/><a NAME="E5:21_6_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A61</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c17">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="c11">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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="c11">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">q9</font>,<font color="Maroon" title="c17">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="E6:21_6_1_1"/><span class="lab"><font color="Green" title="E30">A62</font></span>: 
<span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">q9</font>,<font color="Maroon" title="c17">u9</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattice8.html#D4" target="_self" title="LATTICE8:def.4">Def5</a></span>;<br/><a NAME="E7:21_6_1_1"/>
<span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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="c11">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_6_1_1"><span class="lab"><font color="Green" title="E28">A60</font></span></a>, <a class="ref" href="lattice8.html#D4" target="_self" title="LATTICE8:def.4">Def5</a></span>;<br/><span class="kw">hence </span><a NAME="E8:21_6_1_1"/>
<span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_6_1_1"><span class="lab"><font color="Green" title="E28">A60</font></span></a>, <a class="txt" href="#E6:21_6_1_1"><span class="lab"><font color="Green" title="E30">A62</font></span></a>, <a class="txt" href="#E5:21_6_1_1"><span class="lab"><font color="Green" title="E29">A61</font></span></a>, <a class="ref" href="lattice8.html#D4" target="_self" title="LATTICE8:def.4">Def5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="21_6_1_2"><span class="kw">suppose </span></a><a NAME="E1:21_6_1_2"/><span class="lab"><font color="Green" title="E28">A63</font></span>: 
( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span></span><br/><div class="add"><a NAME="E2:21_6_1_2"/>
<font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c17">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="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c17">u9</font>,<font color="Maroon" title="c16">q9</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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="#E4:21"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="ref" href="lattice5.html#D7" title="LATTICE5:def.7">LATTICE5:def 7</a></span>;<br/><a NAME="E3:21_6_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c17">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="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">q9</font>,<font color="Maroon" title="c17">u9</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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="#E3:21"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="ref" href="lattice5.html#D5" title="LATTICE5:def.5">LATTICE5:def 5</a></span>;<br/><a NAME="E4:21_6_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c17">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="c11">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">q9</font>,<font color="Maroon" title="c17">u9</font>)</span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_1.html#T2" title="WAYBEL_1:th.2">WAYBEL_1:2</a></span>;<br/><a NAME="E5:21_6_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A64</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c17">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="c11">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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="c11">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">q9</font>,<font color="Maroon" title="c17">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="E6:21_6_1_2"/><span class="lab"><font color="Green" title="E30">A65</font></span>: 
<span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">q9</font>,<font color="Maroon" title="c17">u9</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattice8.html#D4" target="_self" title="LATTICE8:def.4">Def5</a></span>;<br/><a NAME="E7:21_6_1_2"/>
<span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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="c11">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_6_1_2"><span class="lab"><font color="Green" title="E28">A63</font></span></a>, <a class="ref" href="lattice8.html#D4" target="_self" title="LATTICE8:def.4">Def5</a></span>;<br/><span class="kw">hence </span><a NAME="E8:21_6_1_2"/>
<span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_6_1_2"><span class="lab"><font color="Green" title="E28">A63</font></span></a>, <a class="txt" href="#E6:21_6_1_2"><span class="lab"><font color="Green" title="E30">A65</font></span></a>, <a class="txt" href="#E5:21_6_1_2"><span class="lab"><font color="Green" title="E29">A64</font></span></a>, <a class="ref" href="lattice8.html#D4" target="_self" title="LATTICE8:def.4">Def5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="E13:21"/><span class="lab"><font color="Green" title="E26">A66</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="lattice8.html#K1" title="LATTICE8:func.1">new_set2</a> <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Olive" title="b2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Olive" title="b3">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> holds <br/><span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Olive" title="b1">p</font>,<font color="Olive" title="b3">u</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</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="21_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">p</font>, <font color="Maroon" title="c14">q</font>, <font color="Maroon" title="c15">u</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice8.html#K1" title="LATTICE8:func.1">new_set2</a> <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> implies <span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:21_7"/><span class="lab"><font color="Green" title="E27">A67</font></span>: 
( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_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="E1:21_7_1"/>
( ( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_7"><span class="lab"><font color="Green" title="E27">A67</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_7_1_1"><span class="kw">suppose </span></a><a NAME="E1:21_7_1_1"/><span class="lab"><font color="Green" title="E28">A68</font></span>: 
( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span></span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c16">p9</font> = <font color="Maroon" title="c13">p</font>, <font color="Maroon" title="c17">u9</font> = <font color="Maroon" title="c15">u</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> ;<br/><a NAME="E3:21_7_1_1"/>
<font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">p9</font>,<font color="Maroon" title="c17">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="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</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="c17">u9</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:21"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="ref" href="lattice5.html#D7" title="LATTICE5:def.7">LATTICE5:def 7</a></span>;<br/><a NAME="E4:21_7_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A69</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c17">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="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c17">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="c11">a</font> &amp; <font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">p9</font>,<font color="Maroon" title="c17">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="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c17">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="#E3:21"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="ref" href="lattice5.html#D5" title="LATTICE5:def.5">LATTICE5:def 5</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="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c17">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="c11">a</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c17">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="c11">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="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c17">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="c11">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">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="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c17">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="c11">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">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="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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="c11">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c17">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="c11">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="E6:21_7_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A70</font></span>: 
<font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">p9</font>,<font color="Maroon" title="c17">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="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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="c11">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c17">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="c11">a</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:21_7_1_1"><span class="lab"><font color="Green" title="E29">A69</font></span></a>, <a class="ref" href="orders_2.html#T3" title="ORDERS_2:th.3">ORDERS_2:3</a></span>;<br/><a NAME="E7:21_7_1_1"/>
( <span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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="c11">a</font> &amp; <span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c17">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="c11">a</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_7_1_1"><span class="lab"><font color="Green" title="E28">A68</font></span></a>, <a class="ref" href="lattice8.html#D4" target="_self" title="LATTICE8:def.4">Def5</a></span>;<br/><span class="kw">hence </span><a NAME="E8:21_7_1_1"/>
<span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:21_7_1_1"><span class="lab"><font color="Green" title="E30">A70</font></span></a>, <a class="ref" href="lattice8.html#D4" target="_self" title="LATTICE8:def.4">Def5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="21_7_1_2"><span class="kw">suppose </span></a><a NAME="E1:21_7_1_2"/><span class="lab"><font color="Green" title="E28">A71</font></span>: 
( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span></span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c16">p9</font> = <font color="Maroon" title="c13">p</font>, <font color="Maroon" title="c17">u9</font> = <font color="Maroon" title="c15">u</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> ;<br/><a NAME="E3:21_7_1_2"/>
<font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">p9</font>,<font color="Maroon" title="c17">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="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</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="c17">u9</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:21"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="ref" href="lattice5.html#D7" title="LATTICE5:def.7">LATTICE5:def 7</a></span>;<br/><a NAME="E4:21_7_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A72</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c17">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="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c17">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="c11">a</font> &amp; <font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">p9</font>,<font color="Maroon" title="c17">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="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c17">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="#E3:21"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="ref" href="lattice5.html#D5" title="LATTICE5:def.5">LATTICE5:def 5</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="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c17">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="c11">a</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c17">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="c11">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="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c17">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="c11">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">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="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c17">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="c11">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">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="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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="c11">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c17">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="c11">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="E6:21_7_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A73</font></span>: 
<font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">p9</font>,<font color="Maroon" title="c17">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="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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="c11">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c17">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="c11">a</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:21_7_1_2"><span class="lab"><font color="Green" title="E29">A72</font></span></a>, <a class="ref" href="orders_2.html#T3" title="ORDERS_2:th.3">ORDERS_2:3</a></span>;<br/><a NAME="E7:21_7_1_2"/>
( <span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">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="c11">a</font> &amp; <span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c17">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="c11">a</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_7_1_2"><span class="lab"><font color="Green" title="E28">A71</font></span></a>, <a class="ref" href="lattice8.html#D4" target="_self" title="LATTICE8:def.4">Def5</a></span>;<br/><span class="kw">hence </span><a NAME="E8:21_7_1_2"/>
<span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:21_7_1_2"><span class="lab"><font color="Green" title="E30">A73</font></span></a>, <a class="ref" href="lattice8.html#D4" target="_self" title="LATTICE8:def.4">Def5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="E14:21"/><span class="lab"><font color="Green" title="E27">A74</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="lattice8.html#K1" title="LATTICE8:func.1">new_set2</a> <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Olive" title="b2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Olive" title="b3">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> holds <br/><span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Olive" title="b1">p</font>,<font color="Olive" title="b3">u</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</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="21_8"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">p</font>, <font color="Maroon" title="c14">q</font>, <font color="Maroon" title="c15">u</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice8.html#K1" title="LATTICE8:func.1">new_set2</a> <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> implies <span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:21_8"/>
( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c16">p9</font> = <font color="Maroon" title="c13">p</font>, <font color="Maroon" title="c17">q9</font> = <font color="Maroon" title="c14">q</font>, <font color="Maroon" title="c18">u9</font> = <font color="Maroon" title="c15">u</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> ;<br/>
<a NAME="E3:21_8"/><span class="lab"><font color="Green" title="E28">A75</font></span>: 
<span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c17">q9</font>,<font color="Maroon" title="c18">u9</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice8.html#D4" target="_self" title="LATTICE8:def.4">Def5</a></span>;<br/>
<a NAME="E4:21_8"/>
( <span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">p9</font>,<font color="Maroon" title="c18">u9</font>) &amp; <span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">d</font> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c16">p9</font>,<font color="Maroon" title="c17">q9</font>) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice8.html#D4" target="_self" title="LATTICE8:def.4">Def5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:21_8"/>
<span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:21"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E3:21_8"><span class="lab"><font color="Green" title="E28">A75</font></span></a>, <a class="ref" href="lattice5.html#D7" title="LATTICE5:def.7">LATTICE5:def 7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E15:21"/>
 for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font>, <font color="Olive" title="b3">u</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice8.html#K1" title="LATTICE8:func.1">new_set2</a> <font color="Maroon" title="c1">A</font> holds  <span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Olive" title="b1">p</font>,<font color="Olive" title="b3">u</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</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="21_9"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">p</font>, <font color="Maroon" title="c14">q</font>, <font color="Maroon" title="c15">u</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice8.html#K1" title="LATTICE8:func.1">new_set2</a> <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_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="E1:21_9_1"/>
( ( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> ) or ( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> ) or ( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> ) or ( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> ) or ( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> ) or ( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> ) or ( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> ) or ( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">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="21_9_1_1"><span class="kw">suppose </span></a><a NAME="E1:21_9_1_1"/>
( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:21_9_1_1"/>
<span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:21"><span class="lab"><font color="Green" title="E27">A74</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="21_9_1_2"><span class="kw">suppose </span></a><a NAME="E1:21_9_1_2"/>
( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:21_9_1_2"/>
<span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:21"><span class="lab"><font color="Green" title="E19">A4</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="21_9_1_3"><span class="kw">suppose </span></a><a NAME="E1:21_9_1_3"/>
( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:21_9_1_3"/>
<span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:21"><span class="lab"><font color="Green" title="E26">A66</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="21_9_1_4"><span class="kw">suppose </span></a><a NAME="E1:21_9_1_4"/>
( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:21_9_1_4"/>
<span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:21"><span class="lab"><font color="Green" title="E21">A13</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="21_9_1_5"><span class="kw">suppose </span></a><a NAME="E1:21_9_1_5"/>
( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:21_9_1_5"/>
<span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:21"><span class="lab"><font color="Green" title="E25">A57</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="21_9_1_6"><span class="kw">suppose </span></a><a NAME="E1:21_9_1_6"/>
( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:21_9_1_6"/>
<span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:21"><span class="lab"><font color="Green" title="E22">A25</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="21_9_1_7"><span class="kw">suppose </span></a><a NAME="E1:21_9_1_7"/>
( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:21_9_1_7"/>
<span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:21"><span class="lab"><font color="Green" title="E24">A45</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="21_9_1_8"><span class="kw">suppose </span></a><a NAME="E1:21_9_1_8"/>
( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:21_9_1_8"/>
<span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c15">u</font>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font>)</span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">q</font>)</span>)</span> <a href="lattice5.html#K3" title="LATTICE5:func.3">.</a> (<font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c15">u</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:21"><span class="lab"><font color="Green" title="E23">A35</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="E16:21"/>
 <a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> (<font color="Maroon" title="c4">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#D7" title="LATTICE5:def.7">LATTICE5:def 7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
