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<b>let </b><font color="Maroon" title="c1">A</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">L</font> being <a href="yellow_0.html#V1" title="YELLOW_0:attr.1">lower-bounded</a> <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a><br/> for <font color="Olive" title="b2">d</font> being  <a href="lattice5.html#NM3" title="LATTICE5:NM.3">BiFunction</a> of <font color="Maroon" title="c1">A</font>,<font color="Olive" title="b1">L</font>  st <font color="Olive" title="b2">d</font> is <a href="lattice5.html#V1" title="LATTICE5:attr.1">symmetric</a> &amp; <font color="Olive" title="b2">d</font> is <a href="lattice5.html#V3" title="LATTICE5:attr.3">u.t.i.</a> holds <br/>for <font color="Olive" title="b3">q</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K4" title="ZFMISC_1:func.4">[:</a><span class="default"><font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">L</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">L</font></span><a href="zfmisc_1.html#K4" title="ZFMISC_1:func.4">:]</a></span>  st <font color="Olive" title="b2">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b3">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">q</font> <a href="mcart_1.html#K10" title="MCART_1:func.10">`3</a> </span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">q</font> <a href="mcart_1.html#K11" title="MCART_1:func.11">`4</a> </span>)</span> holds <br/> <a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Olive" title="b2">d</font>,<font color="Olive" title="b3">q</font> is <a href="lattice5.html#V3" title="LATTICE5:attr.3">u.t.i.</a></span><br/><b>let </b><font color="Maroon" title="c2">L</font> be  <a href="yellow_0.html#V1" title="YELLOW_0:attr.1">lower-bounded</a> <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">d</font> being  <a href="lattice5.html#NM3" title="LATTICE5:NM.3">BiFunction</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">L</font>  st <font color="Olive" title="b1">d</font> is <a href="lattice5.html#V1" title="LATTICE5:attr.1">symmetric</a> &amp; <font color="Olive" title="b1">d</font> is <a href="lattice5.html#V3" title="LATTICE5:attr.3">u.t.i.</a> holds <br/>for <font color="Olive" title="b2">q</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K4" title="ZFMISC_1:func.4">[:</a><span class="default"><font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L</font></span><a href="zfmisc_1.html#K4" title="ZFMISC_1:func.4">:]</a></span>  st <font color="Olive" title="b1">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b2">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">q</font> <a href="mcart_1.html#K10" title="MCART_1:func.10">`3</a> </span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">q</font> <a href="mcart_1.html#K11" title="MCART_1:func.11">`4</a> </span>)</span> holds <br/> <a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Olive" title="b1">d</font>,<font color="Olive" title="b2">q</font> is <a href="lattice5.html#V3" title="LATTICE5:attr.3">u.t.i.</a></span><br/><b>let </b><font color="Maroon" title="c3">d</font> be   <a href="lattice5.html#NM3" title="LATTICE5:NM.3">BiFunction</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">d</font> is <a href="lattice5.html#V1" title="LATTICE5:attr.1">symmetric</a> &amp; <font color="Maroon" title="c3">d</font> is <a href="lattice5.html#V3" title="LATTICE5:attr.3">u.t.i.</a> implies for <font color="Olive" title="b1">q</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K4" title="ZFMISC_1:func.4">[:</a><span class="default"><font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L</font></span><a href="zfmisc_1.html#K4" title="ZFMISC_1:func.4">:]</a></span>  st <font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b1">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">q</font> <a href="mcart_1.html#K10" title="MCART_1:func.10">`3</a> </span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">q</font> <a href="mcart_1.html#K11" title="MCART_1:func.11">`4</a> </span>)</span> holds <br/> <a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Olive" title="b1">q</font> is <a href="lattice5.html#V3" title="LATTICE5:attr.3">u.t.i.</a> )</span><br/>





<b>assume </b><b>that </b><br/><a NAME="E1:40"/><i><font color="Green" title="E27">A1</font></i>: 
<font color="Maroon" title="c3">d</font> is <a href="lattice5.html#V1" title="LATTICE5:attr.1">symmetric</a>
 <b>and </b><br/><a NAME="E2:40"/><i><font color="Green" title="E28">A2</font></i>: 
<font color="Maroon" title="c3">d</font> is <a href="lattice5.html#V3" title="LATTICE5:attr.3">u.t.i.</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">q</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K4" title="ZFMISC_1:func.4">[:</a><span class="default"><font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L</font></span><a href="zfmisc_1.html#K4" title="ZFMISC_1:func.4">:]</a></span>  st <font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b1">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">q</font> <a href="mcart_1.html#K10" title="MCART_1:func.10">`3</a> </span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">q</font> <a href="mcart_1.html#K11" title="MCART_1:func.11">`4</a> </span>)</span> holds <br/> <a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Olive" title="b1">q</font> is <a href="lattice5.html#V3" title="LATTICE5:attr.3">u.t.i.</a></span><br/>

<b>let </b><font color="Maroon" title="c4">q</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K4" title="ZFMISC_1:func.4">[:</a><span class="default"><font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L</font></span><a href="zfmisc_1.html#K4" title="ZFMISC_1:func.4">:]</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K10" title="MCART_1:func.10">`3</a> </span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K11" title="MCART_1:func.11">`4</a> </span>)</span> implies  <a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font> is <a href="lattice5.html#V3" title="LATTICE5:attr.3">u.t.i.</a> )</span><br/>

<b>assume </b><a NAME="E3:40"/><i><font color="Green" title="E29">A3</font></i>: 
<font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K10" title="MCART_1:func.10">`3</a> </span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K11" title="MCART_1:func.11">`4</a> </span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font> is <a href="lattice5.html#V3" title="LATTICE5:attr.3">u.t.i.</a></span><br/>

<b>set </b><font color="Maroon" title="c5">x</font> = <font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> ;<br/>
<b>set </b><font color="Maroon" title="c6">y</font> = <font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> ;<br/>
<b>set </b><font color="Maroon" title="c7">f</font> =  <a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">B</font> = <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<b>reconsider </b><font color="Maroon" title="c9">a</font> = <font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K10" title="MCART_1:func.10">`3</a> , <font color="Maroon" title="c10">b</font> = <font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K11" title="MCART_1:func.11">`4</a>  as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L</font> ;<br/>
<a NAME="E6:40"/><i><font color="Green" title="E30">A4</font></i>: 
for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font>, <font color="Olive" title="b3">u</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice5.html#K6" title="LATTICE5:func.6">new_set</a> <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Olive" title="b2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Olive" title="b3">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> holds <br/><span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b3">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Olive" title="b2">q</font>,<font color="Olive" title="b3">u</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">p</font>, <font color="Maroon" title="c12">q</font>, <font color="Maroon" title="c13">u</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice5.html#K6" title="LATTICE5:func.6">new_set</a> <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> implies <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:40_1"/>
( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/>

<b>then reconsider </b><font color="Maroon" title="c14">p'</font> = <font color="Maroon" title="c11">p</font>, <font color="Maroon" title="c15">q'</font> = <font color="Maroon" title="c12">q</font>, <font color="Maroon" title="c16">u'</font> = <font color="Maroon" title="c13">u</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> ;<br/>
<a NAME="E3:40_1"/><i><font color="Green" title="E31">A5</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<font color="Maroon" title="c16">u'</font>
 
<b>by </b><i><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>;<br/>
<a NAME="E4:40_1"/><i><font color="Green" title="E32">A6</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<font color="Maroon" title="c15">q'</font>
 
<b>by </b><i><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>;<br/>
<a NAME="E5:40_1"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c15">q'</font>,<font color="Maroon" title="c16">u'</font>
 
<b>by </b><i><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>;<br/>
<b>hence </b><a NAME="E6:40_1"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:40"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="#E3:40_1"><i><font color="Green" title="E31">A5</font></i></a>, <a class="txt" href="#E4:40_1"><i><font color="Green" title="E32">A6</font></i></a>, <a class="ref" href="lattice5.html#D8" target="_self" title="LATTICE5:def.8">Def8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E7:40"/><i><font color="Green" title="E31">A7</font></i>: 
for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font>, <font color="Olive" title="b3">u</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice5.html#K6" title="LATTICE5:func.6">new_set</a> <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Olive" title="b2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Olive" title="b3">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> holds <br/><span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b3">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Olive" title="b2">q</font>,<font color="Olive" title="b3">u</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">p</font>, <font color="Maroon" title="c12">q</font>, <font color="Maroon" title="c13">u</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice5.html#K6" title="LATTICE5:func.6">new_set</a> <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> implies <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:40_2"/><i><font color="Green" title="E32">A8</font></i>: 
( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:40_2_1"/>
( ( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) )
 </span><b>by </b><i><a class="txt" href="#E1:40_2"><i><font color="Green" title="E32">A8</font></i></a>, <a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_2_1_1"><b>suppose </b></a><a NAME="E1:40_2_1_1"/><i><font color="Green" title="E33">A9</font></i>: 
( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c14">p'</font> = <font color="Maroon" title="c11">p</font>, <font color="Maroon" title="c15">q'</font> = <font color="Maroon" title="c12">q</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> ;<br/><a NAME="E3:40_2_1_1"/><i><font color="Green" title="E34">A10</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font>
 <b>by </b><i><a class="txt" href="#E1:40_2_1_1"><i><font color="Green" title="E33">A9</font></i></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>;<br/><a NAME="E4:40_2_1_1"/><i><font color="Green" title="E35">A11</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c15">q'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font>
 <b>by </b><i><a class="txt" href="#E1:40_2_1_1"><i><font color="Green" title="E33">A9</font></i></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>;<br/><a NAME="E5:40_2_1_1"/><i><font color="Green" title="E36">A12</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<font color="Maroon" title="c15">q'</font>
 <b>by </b><i><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>;<br/><a NAME="E6:40_2_1_1"/>
<font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<font color="Maroon" title="c15">q'</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c15">q'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:40"><i><font color="Green" title="E28">A2</font></i></a>, <a class="ref" href="lattice5.html#D8" target="_self" title="LATTICE5:def.8">Def8</a></i>;<br/><a NAME="E7:40_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<font color="Maroon" title="c15">q'</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c15">q'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font>
 <b>by </b><i><a class="ref" href="waybel_1.html#T3" title="WAYBEL_1:th.3">WAYBEL_1:3</a></i>;<br/><b>hence </b><a NAME="E8:40_2_1_1"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:40_2_1_1"><i><font color="Green" title="E34">A10</font></i></a>, <a class="txt" href="#E4:40_2_1_1"><i><font color="Green" title="E35">A11</font></i></a>, <a class="txt" href="#E5:40_2_1_1"><i><font color="Green" title="E36">A12</font></i></a>, <a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_2_1_2"><b>suppose </b></a><a NAME="E1:40_2_1_2"/><i><font color="Green" title="E33">A13</font></i>: 
( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c14">p'</font> = <font color="Maroon" title="c11">p</font>, <font color="Maroon" title="c15">q'</font> = <font color="Maroon" title="c12">q</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> ;<br/><a NAME="E3:40_2_1_2"/><i><font color="Green" title="E34">A14</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
 <b>by </b><i><a class="txt" href="#E1:40_2_1_2"><i><font color="Green" title="E33">A13</font></i></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>;<br/><a NAME="E4:40_2_1_2"/><i><font color="Green" title="E35">A15</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c15">q'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
 <b>by </b><i><a class="txt" href="#E1:40_2_1_2"><i><font color="Green" title="E33">A13</font></i></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>;<br/><a NAME="E5:40_2_1_2"/><i><font color="Green" title="E36">A16</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<font color="Maroon" title="c15">q'</font>
 <b>by </b><i><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>;<br/><a NAME="E6:40_2_1_2"/>
<font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<font color="Maroon" title="c15">q'</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c15">q'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:40"><i><font color="Green" title="E28">A2</font></i></a>, <a class="ref" href="lattice5.html#D8" target="_self" title="LATTICE5:def.8">Def8</a></i>;<br/><a NAME="E7:40_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<font color="Maroon" title="c15">q'</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c15">q'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font>
 <b>by </b><i><a class="ref" href="waybel_1.html#T3" title="WAYBEL_1:th.3">WAYBEL_1:3</a></i>;<br/><a NAME="E8:40_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<font color="Maroon" title="c15">q'</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c15">q'</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
 <b>by </b><i><a class="ref" href="waybel_1.html#T3" title="WAYBEL_1:th.3">WAYBEL_1:3</a></i>;<br/><a NAME="E9:40_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<font color="Maroon" title="c15">q'</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c15">q'</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
 <b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>;<br/><b>hence </b><a NAME="E10:40_2_1_2"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:40_2_1_2"><i><font color="Green" title="E34">A14</font></i></a>, <a class="txt" href="#E4:40_2_1_2"><i><font color="Green" title="E35">A15</font></i></a>, <a class="txt" href="#E5:40_2_1_2"><i><font color="Green" title="E36">A16</font></i></a>, <a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_2_1_3"><b>suppose </b></a><a NAME="E1:40_2_1_3"/><i><font color="Green" title="E33">A17</font></i>: 
( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c14">p'</font> = <font color="Maroon" title="c11">p</font>, <font color="Maroon" title="c15">q'</font> = <font color="Maroon" title="c12">q</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> ;<br/><a NAME="E3:40_2_1_3"/><i><font color="Green" title="E34">A18</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
 <b>by </b><i><a class="txt" href="#E1:40_2_1_3"><i><font color="Green" title="E33">A17</font></i></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>;<br/><a NAME="E4:40_2_1_3"/><i><font color="Green" title="E35">A19</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c15">q'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
 <b>by </b><i><a class="txt" href="#E1:40_2_1_3"><i><font color="Green" title="E33">A17</font></i></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>;<br/><a NAME="E5:40_2_1_3"/><i><font color="Green" title="E36">A20</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<font color="Maroon" title="c15">q'</font>
 <b>by </b><i><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>;<br/><a NAME="E6:40_2_1_3"/>
<font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<font color="Maroon" title="c15">q'</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c15">q'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:40"><i><font color="Green" title="E28">A2</font></i></a>, <a class="ref" href="lattice5.html#D8" target="_self" title="LATTICE5:def.8">Def8</a></i>;<br/><a NAME="E7:40_2_1_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<font color="Maroon" title="c15">q'</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c15">q'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
 <b>by </b><i><a class="ref" href="waybel_1.html#T3" title="WAYBEL_1:th.3">WAYBEL_1:3</a></i>;<br/><b>hence </b><a NAME="E8:40_2_1_3"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:40_2_1_3"><i><font color="Green" title="E34">A18</font></i></a>, <a class="txt" href="#E4:40_2_1_3"><i><font color="Green" title="E35">A19</font></i></a>, <a class="txt" href="#E5:40_2_1_3"><i><font color="Green" title="E36">A20</font></i></a>, <a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E8:40"/><i><font color="Green" title="E32">A21</font></i>: 
for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font>, <font color="Olive" title="b3">u</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice5.html#K6" title="LATTICE5:func.6">new_set</a> <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Olive" title="b2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> &amp; <font color="Olive" title="b3">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> holds <br/><span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b3">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Olive" title="b2">q</font>,<font color="Olive" title="b3">u</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">p</font>, <font color="Maroon" title="c12">q</font>, <font color="Maroon" title="c13">u</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice5.html#K6" title="LATTICE5:func.6">new_set</a> <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> implies <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:40_3"/><i><font color="Green" title="E33">A22</font></i>: 
( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:40_3_1"/>
( ( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) )
 </span><b>by </b><i><a class="txt" href="#E1:40_3"><i><font color="Green" title="E33">A22</font></i></a>, <a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_3_1_1"><b>suppose </b></a><a NAME="E1:40_3_1_1"/><i><font color="Green" title="E34">A23</font></i>: 
( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c14">p'</font> = <font color="Maroon" title="c11">p</font>, <font color="Maroon" title="c15">u'</font> = <font color="Maroon" title="c13">u</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> ;<br/><a NAME="E3:40_3_1_1"/><i><font color="Green" title="E35">A24</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font>
 <b>by </b><i><a class="txt" href="#E1:40_3_1_1"><i><font color="Green" title="E34">A23</font></i></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>;<br/><a NAME="E4:40_3_1_1"/><i><font color="Green" title="E36">A25</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c15">u'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font>
 <b>by </b><i><a class="txt" href="#E1:40_3_1_1"><i><font color="Green" title="E34">A23</font></i></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>;<br/><a NAME="E5:40_3_1_1"/><i><font color="Green" title="E37">A26</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c15">u'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c15">u'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T22" title="YELLOW_0:th.22">YELLOW_0:22</a></i>;<br/><i><font color="Green" title="E38">A27</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c15">u'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c15">u'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c15">u'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="yellow_5.html#T1" title="YELLOW_5:th.1">YELLOW_5:1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c15">u'</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c15">u'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>
;<br/><a NAME="E7:40_3_1_1"/>
<font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">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"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<font color="Maroon" title="c15">u'</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:40"><i><font color="Green" title="E28">A2</font></i></a>, <a class="ref" href="lattice5.html#D8" target="_self" title="LATTICE5:def.8">Def8</a></i>;<br/><a NAME="E8:40_3_1_1"/><b>then </b>
<font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">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"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c15">u'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:40"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="lattice5.html#D6" target="_self" title="LATTICE5:def.6">Def6</a></i>;<br/><a NAME="E9:40_3_1_1"/><b>then </b>
<font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">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"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c15">u'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E5:40_3_1_1"><i><font color="Green" title="E37">A26</font></i></a>, <a class="txt" href="#E6:40_3_1_1"><i><font color="Green" title="E38">A27</font></i></a>, <a class="ref" href="orders_2.html#T26" title="ORDERS_2:th.26">ORDERS_2:26</a></i>;<br/><b>hence </b><a NAME="E10:40_3_1_1"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:40_3_1_1"><i><font color="Green" title="E35">A24</font></i></a>, <a class="txt" href="#E4:40_3_1_1"><i><font color="Green" title="E36">A25</font></i></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_3_1_2"><b>suppose </b></a><a NAME="E1:40_3_1_2"/><i><font color="Green" title="E34">A28</font></i>: 
( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c14">p'</font> = <font color="Maroon" title="c11">p</font>, <font color="Maroon" title="c15">u'</font> = <font color="Maroon" title="c13">u</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> ;<br/><a NAME="E3:40_3_1_2"/><i><font color="Green" title="E35">A29</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
 <b>by </b><i><a class="txt" href="#E1:40_3_1_2"><i><font color="Green" title="E34">A28</font></i></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>;<br/><a NAME="E4:40_3_1_2"/><i><font color="Green" title="E36">A30</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c15">u'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
 <b>by </b><i><a class="txt" href="#E1:40_3_1_2"><i><font color="Green" title="E34">A28</font></i></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>;<br/><a NAME="E5:40_3_1_2"/><i><font color="Green" title="E37">A31</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c15">u'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c15">u'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_0.html#T22" title="YELLOW_0:th.22">YELLOW_0:22</a></i>;<br/><i><font color="Green" title="E38">A32</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c15">u'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c15">u'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c15">u'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="yellow_5.html#T1" title="YELLOW_5:th.1">YELLOW_5:1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c15">u'</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c15">u'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c15">u'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c15">u'</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>
;<br/><a NAME="E7:40_3_1_2"/>
<font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">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"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<font color="Maroon" title="c15">u'</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:40"><i><font color="Green" title="E28">A2</font></i></a>, <a class="ref" href="lattice5.html#D8" target="_self" title="LATTICE5:def.8">Def8</a></i>;<br/><a NAME="E8:40_3_1_2"/><b>then </b>
<font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">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"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c15">u'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:40"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="lattice5.html#D6" target="_self" title="LATTICE5:def.6">Def6</a></i>;<br/><a NAME="E9:40_3_1_2"/><b>then </b>
<font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">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"><span class="p3">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c15">u'</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E5:40_3_1_2"><i><font color="Green" title="E37">A31</font></i></a>, <a class="txt" href="#E6:40_3_1_2"><i><font color="Green" title="E38">A32</font></i></a>, <a class="ref" href="orders_2.html#T26" title="ORDERS_2:th.26">ORDERS_2:26</a></i>;<br/><b>hence </b><a NAME="E10:40_3_1_2"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:40_3_1_2"><i><font color="Green" title="E35">A29</font></i></a>, <a class="txt" href="#E4:40_3_1_2"><i><font color="Green" title="E36">A30</font></i></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_3_1_3"><b>suppose </b></a><a NAME="E1:40_3_1_3"/><i><font color="Green" title="E34">A33</font></i>: 
( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c14">p'</font> = <font color="Maroon" title="c11">p</font>, <font color="Maroon" title="c15">u'</font> = <font color="Maroon" title="c13">u</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> ;<br/><a NAME="E3:40_3_1_3"/><i><font color="Green" title="E35">A34</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
 <b>by </b><i><a class="txt" href="#E1:40_3_1_3"><i><font color="Green" title="E34">A33</font></i></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>;<br/><a NAME="E4:40_3_1_3"/><i><font color="Green" title="E36">A35</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c15">u'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
 <b>by </b><i><a class="txt" href="#E1:40_3_1_3"><i><font color="Green" title="E34">A33</font></i></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>;<br/><a NAME="E5:40_3_1_3"/><i><font color="Green" title="E37">A36</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c15">u'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c15">u'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T22" title="YELLOW_0:th.22">YELLOW_0:22</a></i>;<br/><i><font color="Green" title="E38">A37</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c15">u'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c15">u'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c15">u'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="yellow_5.html#T1" title="YELLOW_5:th.1">YELLOW_5:1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c15">u'</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c15">u'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>
;<br/><a NAME="E7:40_3_1_3"/>
<font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">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"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span>,<font color="Maroon" title="c15">u'</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:40"><i><font color="Green" title="E28">A2</font></i></a>, <a class="ref" href="lattice5.html#D8" target="_self" title="LATTICE5:def.8">Def8</a></i>;<br/><a NAME="E8:40_3_1_3"/><b>then </b>
<font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">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"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c15">u'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:40"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="lattice5.html#D6" target="_self" title="LATTICE5:def.6">Def6</a></i>;<br/><a NAME="E9:40_3_1_3"/><b>then </b>
<font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">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"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c15">u'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E5:40_3_1_3"><i><font color="Green" title="E37">A36</font></i></a>, <a class="txt" href="#E6:40_3_1_3"><i><font color="Green" title="E38">A37</font></i></a>, <a class="ref" href="orders_2.html#T26" title="ORDERS_2:th.26">ORDERS_2:26</a></i>;<br/><b>hence </b><a NAME="E10:40_3_1_3"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:40_3_1_3"><i><font color="Green" title="E35">A34</font></i></a>, <a class="txt" href="#E4:40_3_1_3"><i><font color="Green" title="E36">A35</font></i></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E9:40"/><i><font color="Green" title="E33">A38</font></i>: 
for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font>, <font color="Olive" title="b3">u</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice5.html#K6" title="LATTICE5:func.6">new_set</a> <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Olive" title="b2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">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="c8">B</font> holds <br/><span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b3">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Olive" title="b2">q</font>,<font color="Olive" title="b3">u</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">p</font>, <font color="Maroon" title="c12">q</font>, <font color="Maroon" title="c13">u</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice5.html#K6" title="LATTICE5:func.6">new_set</a> <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> implies <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:40_4"/><i><font color="Green" title="E34">A39</font></i>: 
( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/>

<b>then reconsider </b><font color="Maroon" title="c14">p'</font> = <font color="Maroon" title="c11">p</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> ;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:40_4_1"/>
( ( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) )
 </span><b>by </b><i><a class="txt" href="#E1:40_4"><i><font color="Green" title="E34">A39</font></i></a>, <a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_4_1_1"><b>suppose </b></a><a NAME="E1:40_4_1_1"/><i><font color="Green" title="E35">A40</font></i>: 
( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font>
<b>by </b><i><a class="ref" href="waybel_1.html#T4" title="WAYBEL_1:th.4">WAYBEL_1:4</a></i>
;<br/><b>hence </b><a NAME="E3:40_4_1_1"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:40_4_1_1"><i><font color="Green" title="E35">A40</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_4_1_2"><b>suppose </b></a><a NAME="E1:40_4_1_2"/><i><font color="Green" title="E35">A41</font></i>: 
( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><b>then </b><span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
<b>by </b><i><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
<b>by </b><i><a class="txt" href="#E1:40_4_1_2"><i><font color="Green" title="E35">A41</font></i></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>
;<br/><b>hence </b><a NAME="E3:40_4_1_2"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:40_4_1_2"><i><font color="Green" title="E35">A41</font></i></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_4_1_3"><b>suppose </b></a><a NAME="E1:40_4_1_3"/><i><font color="Green" title="E35">A42</font></i>: 
( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><a NAME="E2:40_4_1_3"/><b>then </b><i><font color="Green" title="E36">A43</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font>
 <b>by </b><i><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>;<br/><a NAME="E3:40_4_1_3"/>
<font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:40"><i><font color="Green" title="E28">A2</font></i></a>, <a class="ref" href="lattice5.html#D8" target="_self" title="LATTICE5:def.8">Def8</a></i>;<br/><a NAME="E4:40_4_1_3"/><b>then </b><i><font color="Green" title="E37">A44</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
 <b>by </b><i><a class="ref" href="waybel_1.html#T3" title="WAYBEL_1:th.3">WAYBEL_1:3</a></i>;<br/><a NAME="E5:40_4_1_3"/><i><font color="Green" title="E38">A45</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>;<br/><a NAME="E6:40_4_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
 ;<br/><a NAME="E7:40_4_1_3"/><b>then </b><i><font color="Green" title="E39">A46</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:40"><i><font color="Green" title="E29">A3</font></i></a>, <a class="ref" href="yellow_3.html#T3" title="YELLOW_3:th.3">YELLOW_3:3</a></i>;<br/><a NAME="E8:40_4_1_3"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
 <b>by </b><i><a class="txt" href="#E1:40_4_1_3"><i><font color="Green" title="E35">A42</font></i></a>, <a class="txt" href="#E4:40_4_1_3"><i><font color="Green" title="E37">A44</font></i></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>;<br/><a NAME="E9:40_4_1_3"/><b>then </b><i><font color="Green" title="E40">A47</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E5:40_4_1_3"><i><font color="Green" title="E38">A45</font></i></a>, <a class="txt" href="#E7:40_4_1_3"><i><font color="Green" title="E39">A46</font></i></a>, <a class="ref" href="orders_2.html#T26" title="ORDERS_2:th.26">ORDERS_2:26</a></i>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
<b>by </b><i><a class="ref" href="yellow_5.html#T1" title="YELLOW_5:th.1">YELLOW_5:1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
<b>by </b><i><a class="ref" href="yellow_5.html#T1" title="YELLOW_5:th.1">YELLOW_5:1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>
;<br/><b>hence </b><a NAME="E11:40_4_1_3"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:40_4_1_3"><i><font color="Green" title="E35">A42</font></i></a>, <a class="txt" href="#E2:40_4_1_3"><i><font color="Green" title="E36">A43</font></i></a>, <a class="txt" href="#E9:40_4_1_3"><i><font color="Green" title="E40">A47</font></i></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_4_1_4"><b>suppose </b></a><a NAME="E1:40_4_1_4"/><i><font color="Green" title="E35">A48</font></i>: 
( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><b>then </b><span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
<b>by </b><i><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
<b>by </b><i><a class="txt" href="#E1:40_4_1_4"><i><font color="Green" title="E35">A48</font></i></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>
;<br/><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
<b>by </b><i><a class="txt" href="#E1:40_4_1_4"><i><font color="Green" title="E35">A48</font></i></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
<b>by </b><i><a class="ref" href="yellow_5.html#T1" title="YELLOW_5:th.1">YELLOW_5:1</a></i>
;<br/><b>hence </b><a NAME="E4:40_4_1_4"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_0.html#T22" title="YELLOW_0:th.22">YELLOW_0:22</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_4_1_5"><b>suppose </b></a><a NAME="E1:40_4_1_5"/><i><font color="Green" title="E35">A49</font></i>: 
( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font>
<b>by </b><i><a class="ref" href="waybel_1.html#T4" title="WAYBEL_1:th.4">WAYBEL_1:4</a></i>
;<br/><b>hence </b><a NAME="E3:40_4_1_5"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:40_4_1_5"><i><font color="Green" title="E35">A49</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_4_1_6"><b>suppose </b></a><a NAME="E1:40_4_1_6"/><i><font color="Green" title="E35">A50</font></i>: 
( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><a NAME="E2:40_4_1_6"/><b>then </b><i><font color="Green" title="E36">A51</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
 <b>by </b><i><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>;<br/><a NAME="E3:40_4_1_6"/>
<font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:40"><i><font color="Green" title="E28">A2</font></i></a>, <a class="ref" href="lattice5.html#D8" target="_self" title="LATTICE5:def.8">Def8</a></i>;<br/><a NAME="E4:40_4_1_6"/><b>then </b><i><font color="Green" title="E37">A52</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
 <b>by </b><i><a class="ref" href="waybel_1.html#T3" title="WAYBEL_1:th.3">WAYBEL_1:3</a></i>;<br/><a NAME="E5:40_4_1_6"/><i><font color="Green" title="E38">A53</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>;<br/><a NAME="E6:40_4_1_6"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
 ;<br/><a NAME="E7:40_4_1_6"/><b>then </b><i><font color="Green" title="E39">A54</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:40"><i><font color="Green" title="E29">A3</font></i></a>, <a class="ref" href="yellow_3.html#T3" title="YELLOW_3:th.3">YELLOW_3:3</a></i>;<br/><a NAME="E8:40_4_1_6"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
 <b>by </b><i><a class="txt" href="#E1:40_4_1_6"><i><font color="Green" title="E35">A50</font></i></a>, <a class="txt" href="#E4:40_4_1_6"><i><font color="Green" title="E37">A52</font></i></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>;<br/><a NAME="E9:40_4_1_6"/><b>then </b><i><font color="Green" title="E40">A55</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E5:40_4_1_6"><i><font color="Green" title="E38">A53</font></i></a>, <a class="txt" href="#E7:40_4_1_6"><i><font color="Green" title="E39">A54</font></i></a>, <a class="ref" href="orders_2.html#T26" title="ORDERS_2:th.26">ORDERS_2:26</a></i>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
<b>by </b><i><a class="ref" href="yellow_5.html#T1" title="YELLOW_5:th.1">YELLOW_5:1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
<b>by </b><i><a class="ref" href="yellow_5.html#T1" title="YELLOW_5:th.1">YELLOW_5:1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font>
<b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font>
<b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>
;<br/><b>hence </b><a NAME="E11:40_4_1_6"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:40_4_1_6"><i><font color="Green" title="E35">A50</font></i></a>, <a class="txt" href="#E2:40_4_1_6"><i><font color="Green" title="E36">A51</font></i></a>, <a class="txt" href="#E9:40_4_1_6"><i><font color="Green" title="E40">A55</font></i></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_4_1_7"><b>suppose </b></a><a NAME="E1:40_4_1_7"/><i><font color="Green" title="E35">A56</font></i>: 
( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><a NAME="E2:40_4_1_7"/><b>then </b><i><font color="Green" title="E36">A57</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
 <b>by </b><i><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>;<br/><a NAME="E3:40_4_1_7"/>
<font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:40"><i><font color="Green" title="E28">A2</font></i></a>, <a class="ref" href="lattice5.html#D8" target="_self" title="LATTICE5:def.8">Def8</a></i>;<br/><a NAME="E4:40_4_1_7"/><b>then </b><i><font color="Green" title="E37">A58</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font>
 <b>by </b><i><a class="ref" href="waybel_1.html#T3" title="WAYBEL_1:th.3">WAYBEL_1:3</a></i>;<br/><a NAME="E5:40_4_1_7"/><i><font color="Green" title="E38">A59</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>;<br/><a NAME="E6:40_4_1_7"/><i><font color="Green" title="E39">A60</font></i>: 
<font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
 <b>by </b><i><a class="txt" href="#E1:40"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="#E3:40"><i><font color="Green" title="E29">A3</font></i></a>, <a class="ref" href="lattice5.html#D6" target="_self" title="LATTICE5:def.6">Def6</a></i>;<br/><a NAME="E7:40_4_1_7"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font>
 ;<br/><a NAME="E8:40_4_1_7"/><b>then </b><i><font color="Green" title="E40">A61</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E6:40_4_1_7"><i><font color="Green" title="E39">A60</font></i></a>, <a class="ref" href="yellow_3.html#T3" title="YELLOW_3:th.3">YELLOW_3:3</a></i>;<br/><a NAME="E9:40_4_1_7"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font>
 <b>by </b><i><a class="txt" href="#E1:40_4_1_7"><i><font color="Green" title="E35">A56</font></i></a>, <a class="txt" href="#E4:40_4_1_7"><i><font color="Green" title="E37">A58</font></i></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>;<br/><a NAME="E10:40_4_1_7"/><b>then </b><i><font color="Green" title="E41">A62</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E5:40_4_1_7"><i><font color="Green" title="E38">A59</font></i></a>, <a class="txt" href="#E8:40_4_1_7"><i><font color="Green" title="E40">A61</font></i></a>, <a class="ref" href="orders_2.html#T26" title="ORDERS_2:th.26">ORDERS_2:26</a></i>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
<b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
<b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
<b>by </b><i><a class="ref" href="yellow_5.html#T1" title="YELLOW_5:th.1">YELLOW_5:1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="yellow_5.html#T1" title="YELLOW_5:th.1">YELLOW_5:1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>
;<br/><b>hence </b><a NAME="E12:40_4_1_7"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:40_4_1_7"><i><font color="Green" title="E35">A56</font></i></a>, <a class="txt" href="#E2:40_4_1_7"><i><font color="Green" title="E36">A57</font></i></a>, <a class="txt" href="#E10:40_4_1_7"><i><font color="Green" title="E41">A62</font></i></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_4_1_8"><b>suppose </b></a><a NAME="E1:40_4_1_8"/><i><font color="Green" title="E35">A63</font></i>: 
( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><a NAME="E2:40_4_1_8"/><b>then </b><i><font color="Green" title="E36">A64</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
 <b>by </b><i><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>;<br/><a NAME="E3:40_4_1_8"/><i><font color="Green" title="E37">A65</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
 <b>by </b><i><a class="txt" href="#E1:40_4_1_8"><i><font color="Green" title="E35">A63</font></i></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>;<br/><a NAME="E4:40_4_1_8"/>
<font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:40"><i><font color="Green" title="E28">A2</font></i></a>, <a class="ref" href="lattice5.html#D8" target="_self" title="LATTICE5:def.8">Def8</a></i>;<br/><a NAME="E5:40_4_1_8"/><b>then </b><i><font color="Green" title="E38">A66</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
 <b>by </b><i><a class="ref" href="waybel_1.html#T3" title="WAYBEL_1:th.3">WAYBEL_1:3</a></i>;<br/><i><font color="Green" title="E39">A67</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="yellow_5.html#T1" title="YELLOW_5:th.1">YELLOW_5:1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="yellow_5.html#T1" title="YELLOW_5:th.1">YELLOW_5:1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>
;<br/><a NAME="E7:40_4_1_8"/><i><font color="Green" title="E40">A68</font></i>: 
<font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
 <b>by </b><i><a class="txt" href="#E1:40"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="#E3:40"><i><font color="Green" title="E29">A3</font></i></a>, <a class="ref" href="lattice5.html#D6" target="_self" title="LATTICE5:def.6">Def6</a></i>;<br/><a NAME="E8:40_4_1_8"/><i><font color="Green" title="E41">A69</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>;<br/><a NAME="E9:40_4_1_8"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
 ;<br/><a NAME="E10:40_4_1_8"/><b>then </b><i><font color="Green" title="E42">A70</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E7:40_4_1_8"><i><font color="Green" title="E40">A68</font></i></a>, <a class="txt" href="#E8:40_4_1_8"><i><font color="Green" title="E41">A69</font></i></a>, <a class="ref" href="yellow_3.html#T3" title="YELLOW_3:th.3">YELLOW_3:3</a></i>;<br/><a NAME="E11:40_4_1_8"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:40_4_1_8"><i><font color="Green" title="E36">A64</font></i></a>, <a class="txt" href="#E5:40_4_1_8"><i><font color="Green" title="E38">A66</font></i></a>, <a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>;<br/><a NAME="E12:40_4_1_8"/><b>then </b><i><font color="Green" title="E43">A71</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E6:40_4_1_8"><i><font color="Green" title="E39">A67</font></i></a>, <a class="txt" href="#E10:40_4_1_8"><i><font color="Green" title="E42">A70</font></i></a>, <a class="ref" href="orders_2.html#T26" title="ORDERS_2:th.26">ORDERS_2:26</a></i>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
<b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
<b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
<b>by </b><i><a class="ref" href="yellow_5.html#T1" title="YELLOW_5:th.1">YELLOW_5:1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">p'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font>
<b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>
;<br/><b>hence </b><a NAME="E14:40_4_1_8"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:40_4_1_8"><i><font color="Green" title="E35">A63</font></i></a>, <a class="txt" href="#E3:40_4_1_8"><i><font color="Green" title="E37">A65</font></i></a>, <a class="txt" href="#E12:40_4_1_8"><i><font color="Green" title="E43">A71</font></i></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_4_1_9"><b>suppose </b></a><a NAME="E1:40_4_1_9"/><i><font color="Green" title="E35">A72</font></i>: 
( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font>
<b>by </b><i><a class="ref" href="waybel_1.html#T4" title="WAYBEL_1:th.4">WAYBEL_1:4</a></i>
;<br/><b>hence </b><a NAME="E3:40_4_1_9"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:40_4_1_9"><i><font color="Green" title="E35">A72</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E10:40"/><i><font color="Green" title="E34">A73</font></i>: 
for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font>, <font color="Olive" title="b3">u</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice5.html#K6" title="LATTICE5:func.6">new_set</a> <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> &amp; <font color="Olive" title="b2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Olive" title="b3">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> holds <br/><span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b3">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Olive" title="b2">q</font>,<font color="Olive" title="b3">u</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">p</font>, <font color="Maroon" title="c12">q</font>, <font color="Maroon" title="c13">u</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice5.html#K6" title="LATTICE5:func.6">new_set</a> <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> implies <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:40_5"/><i><font color="Green" title="E35">A74</font></i>: 
( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/>

<b>then reconsider </b><font color="Maroon" title="c14">q'</font> = <font color="Maroon" title="c12">q</font>, <font color="Maroon" title="c15">u'</font> = <font color="Maroon" title="c13">u</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> ;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_5_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:40_5_1"/>
( ( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> ) or ( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> ) or ( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> ) )
 </span><b>by </b><i><a class="txt" href="#E1:40_5"><i><font color="Green" title="E35">A74</font></i></a>, <a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_5_1_1"><b>suppose </b></a><a NAME="E1:40_5_1_1"/><i><font color="Green" title="E36">A75</font></i>: 
( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><a NAME="E2:40_5_1_1"/><b>then </b><i><font color="Green" title="E37">A76</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font>
 <b>by </b><i><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>;<br/><a NAME="E3:40_5_1_1"/><i><font color="Green" title="E38">A77</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q'</font>,<font color="Maroon" title="c15">u'</font>
 <b>by </b><i><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>;<br/><a NAME="E4:40_5_1_1"/>
<font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c15">u'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c15">u'</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"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:40"><i><font color="Green" title="E28">A2</font></i></a>, <a class="ref" href="lattice5.html#D8" target="_self" title="LATTICE5:def.8">Def8</a></i>;<br/><a NAME="E5:40_5_1_1"/><b>then </b>
<font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c15">u'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q'</font>,<font color="Maroon" title="c15">u'</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:40"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="lattice5.html#D6" target="_self" title="LATTICE5:def.6">Def6</a></i>;<br/><a NAME="E6:40_5_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c15">u'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q'</font>,<font color="Maroon" title="c15">u'</font></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font>
 <b>by </b><i><a class="ref" href="waybel_1.html#T3" title="WAYBEL_1:th.3">WAYBEL_1:3</a></i>;<br/><a NAME="E7:40_5_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c15">u'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q'</font>,<font color="Maroon" title="c15">u'</font></span>)</span>
 <b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>;<br/><b>hence </b><a NAME="E8:40_5_1_1"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:40_5_1_1"><i><font color="Green" title="E36">A75</font></i></a>, <a class="txt" href="#E2:40_5_1_1"><i><font color="Green" title="E37">A76</font></i></a>, <a class="txt" href="#E3:40_5_1_1"><i><font color="Green" title="E38">A77</font></i></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_5_1_2"><b>suppose </b></a><a NAME="E1:40_5_1_2"/><i><font color="Green" title="E36">A78</font></i>: 
( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><a NAME="E2:40_5_1_2"/><b>then </b><i><font color="Green" title="E37">A79</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
 <b>by </b><i><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>;<br/><a NAME="E3:40_5_1_2"/><i><font color="Green" title="E38">A80</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q'</font>,<font color="Maroon" title="c15">u'</font>
 <b>by </b><i><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>;<br/><a NAME="E4:40_5_1_2"/>
<font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c15">u'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c15">u'</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"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:40"><i><font color="Green" title="E28">A2</font></i></a>, <a class="ref" href="lattice5.html#D8" target="_self" title="LATTICE5:def.8">Def8</a></i>;<br/><a NAME="E5:40_5_1_2"/><b>then </b>
<font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c15">u'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q'</font>,<font color="Maroon" title="c15">u'</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:40"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="lattice5.html#D6" target="_self" title="LATTICE5:def.6">Def6</a></i>;<br/><a NAME="E6:40_5_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c15">u'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q'</font>,<font color="Maroon" title="c15">u'</font></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
 <b>by </b><i><a class="ref" href="waybel_1.html#T3" title="WAYBEL_1:th.3">WAYBEL_1:3</a></i>;<br/><a NAME="E7:40_5_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c15">u'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q'</font>,<font color="Maroon" title="c15">u'</font></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
 <b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>;<br/><a NAME="E8:40_5_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c15">u'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q'</font>,<font color="Maroon" title="c15">u'</font></span>)</span>
 <b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>;<br/><a NAME="E9:40_5_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c15">u'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q'</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q'</font>,<font color="Maroon" title="c15">u'</font></span>)</span>
 <b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>;<br/><b>hence </b><a NAME="E10:40_5_1_2"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:40_5_1_2"><i><font color="Green" title="E36">A78</font></i></a>, <a class="txt" href="#E2:40_5_1_2"><i><font color="Green" title="E37">A79</font></i></a>, <a class="txt" href="#E3:40_5_1_2"><i><font color="Green" title="E38">A80</font></i></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_5_1_3"><b>suppose </b></a><a NAME="E1:40_5_1_3"/><i><font color="Green" title="E36">A81</font></i>: 
( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><a NAME="E2:40_5_1_3"/><b>then </b><i><font color="Green" title="E37">A82</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
 <b>by </b><i><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>;<br/><a NAME="E3:40_5_1_3"/><i><font color="Green" title="E38">A83</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q'</font>,<font color="Maroon" title="c15">u'</font>
 <b>by </b><i><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>;<br/><a NAME="E4:40_5_1_3"/>
<font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c15">u'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c15">u'</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"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:40"><i><font color="Green" title="E28">A2</font></i></a>, <a class="ref" href="lattice5.html#D8" target="_self" title="LATTICE5:def.8">Def8</a></i>;<br/><a NAME="E5:40_5_1_3"/><b>then </b>
<font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c15">u'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q'</font>,<font color="Maroon" title="c15">u'</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:40"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="lattice5.html#D6" target="_self" title="LATTICE5:def.6">Def6</a></i>;<br/><a NAME="E6:40_5_1_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c15">u'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q'</font>,<font color="Maroon" title="c15">u'</font></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
 <b>by </b><i><a class="ref" href="waybel_1.html#T3" title="WAYBEL_1:th.3">WAYBEL_1:3</a></i>;<br/><a NAME="E7:40_5_1_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c15">u'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q'</font>,<font color="Maroon" title="c15">u'</font></span>)</span>
 <b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>;<br/><b>hence </b><a NAME="E8:40_5_1_3"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:40_5_1_3"><i><font color="Green" title="E36">A81</font></i></a>, <a class="txt" href="#E2:40_5_1_3"><i><font color="Green" title="E37">A82</font></i></a>, <a class="txt" href="#E3:40_5_1_3"><i><font color="Green" title="E38">A83</font></i></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E11:40"/><i><font color="Green" title="E35">A84</font></i>: 
for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font>, <font color="Olive" title="b3">u</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice5.html#K6" title="LATTICE5:func.6">new_set</a> <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">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="c8">B</font> holds <br/><span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b3">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Olive" title="b2">q</font>,<font color="Olive" title="b3">u</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">p</font>, <font color="Maroon" title="c12">q</font>, <font color="Maroon" title="c13">u</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice5.html#K6" title="LATTICE5:func.6">new_set</a> <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> implies <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:40_6"/><i><font color="Green" title="E36">A85</font></i>: 
( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/>

<b>then reconsider </b><font color="Maroon" title="c14">q'</font> = <font color="Maroon" title="c12">q</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> ;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_6_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:40_6_1"/>
( ( <font color="Maroon" title="c12">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c12">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c12">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c12">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c12">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c12">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c12">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c12">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c12">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) )
 </span><b>by </b><i><a class="txt" href="#E1:40_6"><i><font color="Green" title="E36">A85</font></i></a>, <a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_6_1_1"><b>suppose </b></a><a NAME="E1:40_6_1_1"/>
( <font color="Maroon" title="c12">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><a NAME="E2:40_6_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c2">L</font>
 <b>by </b><i><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>;<br/><b>hence </b><a NAME="E3:40_6_1_1"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_0.html#T44" title="YELLOW_0:th.44">YELLOW_0:44</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_6_1_2"><b>suppose </b></a><a NAME="E1:40_6_1_2"/><i><font color="Green" title="E37">A86</font></i>: 
( <font color="Maroon" title="c12">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><a NAME="E2:40_6_1_2"/><b>then </b><i><font color="Green" title="E38">A87</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b</font>
 <b>by </b><i><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q'</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:40_6_1_2"><i><font color="Green" title="E37">A86</font></i></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q'</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
<b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>
;<br/><b>hence </b><a NAME="E4:40_6_1_2"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:40_6_1_2"><i><font color="Green" title="E38">A87</font></i></a>, <a class="ref" href="yellow_0.html#T22" title="YELLOW_0:th.22">YELLOW_0:22</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_6_1_3"><b>suppose </b></a><a NAME="E1:40_6_1_3"/><i><font color="Green" title="E37">A88</font></i>: 
( <font color="Maroon" title="c12">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><a NAME="E2:40_6_1_3"/><b>then </b><i><font color="Green" title="E38">A89</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
 <b>by </b><i><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:40_6_1_3"><i><font color="Green" title="E37">A88</font></i></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:40_6_1_3"><i><font color="Green" title="E37">A88</font></i></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q'</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>
;<br/><b>hence </b><a NAME="E4:40_6_1_3"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:40_6_1_3"><i><font color="Green" title="E38">A89</font></i></a>, <a class="ref" href="yellow_0.html#T22" title="YELLOW_0:th.22">YELLOW_0:22</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_6_1_4"><b>suppose </b></a><a NAME="E1:40_6_1_4"/><i><font color="Green" title="E37">A90</font></i>: 
( <font color="Maroon" title="c12">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><a NAME="E2:40_6_1_4"/><b>then </b><i><font color="Green" title="E38">A91</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b</font>
 <b>by </b><i><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q'</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:40_6_1_4"><i><font color="Green" title="E37">A90</font></i></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q'</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
<b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>
;<br/><b>hence </b><a NAME="E4:40_6_1_4"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:40_6_1_4"><i><font color="Green" title="E38">A91</font></i></a>, <a class="ref" href="yellow_0.html#T22" title="YELLOW_0:th.22">YELLOW_0:22</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_6_1_5"><b>suppose </b></a><a NAME="E1:40_6_1_5"/>
( <font color="Maroon" title="c12">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><a NAME="E2:40_6_1_5"/><b>then </b>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c2">L</font>
 <b>by </b><i><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>;<br/><b>hence </b><a NAME="E3:40_6_1_5"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_0.html#T44" title="YELLOW_0:th.44">YELLOW_0:44</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_6_1_6"><b>suppose </b></a><a NAME="E1:40_6_1_6"/><i><font color="Green" title="E37">A92</font></i>: 
( <font color="Maroon" title="c12">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><a NAME="E2:40_6_1_6"/><b>then </b><i><font color="Green" title="E38">A93</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a</font>
 <b>by </b><i><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q'</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:40_6_1_6"><i><font color="Green" title="E37">A92</font></i></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q'</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>
<br/>.= 
<font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q'</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>
;<br/><b>hence </b><a NAME="E4:40_6_1_6"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:40_6_1_6"><i><font color="Green" title="E38">A93</font></i></a>, <a class="ref" href="yellow_0.html#T22" title="YELLOW_0:th.22">YELLOW_0:22</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_6_1_7"><b>suppose </b></a><a NAME="E1:40_6_1_7"/><i><font color="Green" title="E37">A94</font></i>: 
( <font color="Maroon" title="c12">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><a NAME="E2:40_6_1_7"/><b>then </b><i><font color="Green" title="E38">A95</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
 <b>by </b><i><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:40_6_1_7"><i><font color="Green" title="E37">A94</font></i></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:40_6_1_7"><i><font color="Green" title="E37">A94</font></i></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q'</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>
;<br/><b>hence </b><a NAME="E4:40_6_1_7"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:40_6_1_7"><i><font color="Green" title="E38">A95</font></i></a>, <a class="ref" href="yellow_0.html#T22" title="YELLOW_0:th.22">YELLOW_0:22</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_6_1_8"><b>suppose </b></a><a NAME="E1:40_6_1_8"/><i><font color="Green" title="E37">A96</font></i>: 
( <font color="Maroon" title="c12">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><a NAME="E2:40_6_1_8"/><b>then </b><i><font color="Green" title="E38">A97</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a</font>
 <b>by </b><i><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q'</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:40_6_1_8"><i><font color="Green" title="E37">A96</font></i></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">q'</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font>
<b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>
;<br/><b>hence </b><a NAME="E4:40_6_1_8"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:40_6_1_8"><i><font color="Green" title="E38">A97</font></i></a>, <a class="ref" href="yellow_0.html#T22" title="YELLOW_0:th.22">YELLOW_0:22</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_6_1_9"><b>suppose </b></a><a NAME="E1:40_6_1_9"/>
( <font color="Maroon" title="c12">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><a NAME="E2:40_6_1_9"/><b>then </b>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c2">L</font>
 <b>by </b><i><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>;<br/><b>hence </b><a NAME="E3:40_6_1_9"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_0.html#T44" title="YELLOW_0:th.44">YELLOW_0:44</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E12:40"/><i><font color="Green" title="E36">A98</font></i>: 
for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font>, <font color="Olive" title="b3">u</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice5.html#K6" title="LATTICE5:func.6">new_set</a> <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">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="c8">B</font> &amp; <font color="Olive" title="b3">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> holds <br/><span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b3">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Olive" title="b2">q</font>,<font color="Olive" title="b3">u</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">p</font>, <font color="Maroon" title="c12">q</font>, <font color="Maroon" title="c13">u</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice5.html#K6" title="LATTICE5:func.6">new_set</a> <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> implies <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:40_7"/><i><font color="Green" title="E37">A99</font></i>: 
( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/>

<b>then reconsider </b><font color="Maroon" title="c14">u'</font> = <font color="Maroon" title="c13">u</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> ;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_7_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:40_7_1"/>
( ( <font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) )
 </span><b>by </b><i><a class="txt" href="#E1:40_7"><i><font color="Green" title="E37">A99</font></i></a>, <a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_7_1_1"><b>suppose </b></a><a NAME="E1:40_7_1_1"/><i><font color="Green" title="E38">A100</font></i>: 
( <font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font>
<b>by </b><i><a class="txt" href="#E1:40_7_1_1"><i><font color="Green" title="E38">A100</font></i></a>, <a class="ref" href="waybel_1.html#T4" title="WAYBEL_1:th.4">WAYBEL_1:4</a></i>
;<br/><b>hence </b><a NAME="E3:40_7_1_1"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_7_1_2"><b>suppose </b></a><a NAME="E1:40_7_1_2"/><i><font color="Green" title="E38">A101</font></i>: 
( <font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span> = 
<font color="Maroon" title="c10">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">u'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
<b>by </b><i><a class="txt" href="#E1:40_7_1_2"><i><font color="Green" title="E38">A101</font></i></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>
;<br/><b>hence </b><a NAME="E3:40_7_1_2"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:40_7_1_2"><i><font color="Green" title="E38">A101</font></i></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_7_1_3"><b>suppose </b></a><a NAME="E1:40_7_1_3"/><i><font color="Green" title="E38">A102</font></i>: 
( <font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><a NAME="E2:40_7_1_3"/><b>then </b><i><font color="Green" title="E39">A103</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">u'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
 <b>by </b><i><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>;<br/><a NAME="E3:40_7_1_3"/>
<font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">u'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">u'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:40"><i><font color="Green" title="E28">A2</font></i></a>, <a class="ref" href="lattice5.html#D8" target="_self" title="LATTICE5:def.8">Def8</a></i>;<br/><a NAME="E4:40_7_1_3"/><b>then </b><i><font color="Green" title="E40">A104</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">u'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">u'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
 <b>by </b><i><a class="ref" href="waybel_1.html#T3" title="WAYBEL_1:th.3">WAYBEL_1:3</a></i>;<br/><a NAME="E5:40_7_1_3"/><i><font color="Green" title="E41">A105</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">u'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">u'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>;<br/><a NAME="E6:40_7_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">u'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">u'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
 ;<br/><a NAME="E7:40_7_1_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">u'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">u'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:40"><i><font color="Green" title="E29">A3</font></i></a>, <a class="ref" href="yellow_3.html#T3" title="YELLOW_3:th.3">YELLOW_3:3</a></i>;<br/><a NAME="E8:40_7_1_3"/><b>then </b><i><font color="Green" title="E42">A106</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">u'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:40_7_1_3"><i><font color="Green" title="E39">A103</font></i></a>, <a class="txt" href="#E4:40_7_1_3"><i><font color="Green" title="E40">A104</font></i></a>, <a class="txt" href="#E5:40_7_1_3"><i><font color="Green" title="E41">A105</font></i></a>, <a class="ref" href="orders_2.html#T26" title="ORDERS_2:th.26">ORDERS_2:26</a></i>;<br/><font color="Maroon" title="c10">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font>
<b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>
<br/>.= 
<font color="Maroon" title="c10">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font>
<b>by </b><i><a class="ref" href="yellow_5.html#T1" title="YELLOW_5:th.1">YELLOW_5:1</a></i>
<br/>.= 
<font color="Maroon" title="c10">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span>
<b>by </b><i><a class="ref" href="yellow_5.html#T1" title="YELLOW_5:th.1">YELLOW_5:1</a></i>
<br/>.= 
<font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>
;<br/><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">u'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">u'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">u'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">u'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:40_7_1_3"><i><font color="Green" title="E38">A102</font></i></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>
;<br/><b>hence </b><a NAME="E11:40_7_1_3"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:40_7_1_3"><i><font color="Green" title="E38">A102</font></i></a>, <a class="txt" href="#E8:40_7_1_3"><i><font color="Green" title="E42">A106</font></i></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_7_1_4"><b>suppose </b></a><a NAME="E1:40_7_1_4"/><i><font color="Green" title="E38">A107</font></i>: 
( <font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span> = 
<font color="Maroon" title="c10">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>
<br/>.= 
<font color="Maroon" title="c10">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">u'</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:40_7_1_4"><i><font color="Green" title="E38">A107</font></i></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>
<br/>.= 
<font color="Maroon" title="c10">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E1:40_7_1_4"><i><font color="Green" title="E38">A107</font></i></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>
<br/>.= 
<font color="Maroon" title="c10">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font></span>)</span>
<b>by </b><i><a class="ref" href="yellow_5.html#T1" title="YELLOW_5:th.1">YELLOW_5:1</a></i>
;<br/><b>hence </b><a NAME="E3:40_7_1_4"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_0.html#T22" title="YELLOW_0:th.22">YELLOW_0:22</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_7_1_5"><b>suppose </b></a><a NAME="E1:40_7_1_5"/><i><font color="Green" title="E38">A108</font></i>: 
( <font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font>
<b>by </b><i><a class="txt" href="#E1:40_7_1_5"><i><font color="Green" title="E38">A108</font></i></a>, <a class="ref" href="waybel_1.html#T4" title="WAYBEL_1:th.4">WAYBEL_1:4</a></i>
;<br/><b>hence </b><a NAME="E3:40_7_1_5"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_7_1_6"><b>suppose </b></a><a NAME="E1:40_7_1_6"/><i><font color="Green" title="E38">A109</font></i>: 
( <font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><a NAME="E2:40_7_1_6"/><b>then </b><i><font color="Green" title="E39">A110</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">u'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
 <b>by </b><i><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>;<br/><a NAME="E3:40_7_1_6"/>
<font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">u'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">u'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:40"><i><font color="Green" title="E28">A2</font></i></a>, <a class="ref" href="lattice5.html#D8" target="_self" title="LATTICE5:def.8">Def8</a></i>;<br/><a NAME="E4:40_7_1_6"/><b>then </b><i><font color="Green" title="E40">A111</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">u'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">u'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
 <b>by </b><i><a class="ref" href="waybel_1.html#T3" title="WAYBEL_1:th.3">WAYBEL_1:3</a></i>;<br/><a NAME="E5:40_7_1_6"/><i><font color="Green" title="E41">A112</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">u'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">u'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>;<br/><i><font color="Green" title="E42">A113</font></i>: <font color="Maroon" title="c10">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font>
<b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>
<br/>.= 
<font color="Maroon" title="c10">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font>
<b>by </b><i><a class="ref" href="yellow_5.html#T1" title="YELLOW_5:th.1">YELLOW_5:1</a></i>
<br/>.= 
<font color="Maroon" title="c10">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span>
<b>by </b><i><a class="ref" href="yellow_5.html#T1" title="YELLOW_5:th.1">YELLOW_5:1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font>
<b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>
;<br/><a NAME="E7:40_7_1_6"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">u'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">u'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
 ;<br/><a NAME="E8:40_7_1_6"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">u'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">u'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:40"><i><font color="Green" title="E29">A3</font></i></a>, <a class="ref" href="yellow_3.html#T3" title="YELLOW_3:th.3">YELLOW_3:3</a></i>;<br/><a NAME="E9:40_7_1_6"/><b>then </b><i><font color="Green" title="E43">A114</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">u'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:40_7_1_6"><i><font color="Green" title="E39">A110</font></i></a>, <a class="txt" href="#E4:40_7_1_6"><i><font color="Green" title="E40">A111</font></i></a>, <a class="txt" href="#E5:40_7_1_6"><i><font color="Green" title="E41">A112</font></i></a>, <a class="ref" href="orders_2.html#T26" title="ORDERS_2:th.26">ORDERS_2:26</a></i>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">u'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">u'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span>
<b>by </b><i><a class="txt" href="#E6:40_7_1_6"><i><font color="Green" title="E42">A113</font></i></a>, <a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">u'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font>
<b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">u'</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font>
<b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">u'</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:40_7_1_6"><i><font color="Green" title="E38">A109</font></i></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>
;<br/><b>hence </b><a NAME="E11:40_7_1_6"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:40_7_1_6"><i><font color="Green" title="E38">A109</font></i></a>, <a class="txt" href="#E9:40_7_1_6"><i><font color="Green" title="E43">A114</font></i></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_7_1_7"><b>suppose </b></a><a NAME="E1:40_7_1_7"/><i><font color="Green" title="E38">A115</font></i>: 
( <font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><a NAME="E2:40_7_1_7"/><b>then </b><i><font color="Green" title="E39">A116</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">u'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font>
 <b>by </b><i><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>;<br/><a NAME="E3:40_7_1_7"/>
<font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">u'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">u'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:40"><i><font color="Green" title="E28">A2</font></i></a>, <a class="ref" href="lattice5.html#D8" target="_self" title="LATTICE5:def.8">Def8</a></i>;<br/><a NAME="E4:40_7_1_7"/><b>then </b><i><font color="Green" title="E40">A117</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">u'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">u'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font>
 <b>by </b><i><a class="ref" href="waybel_1.html#T3" title="WAYBEL_1:th.3">WAYBEL_1:3</a></i>;<br/><i><font color="Green" title="E41">A118</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">u'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">u'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">u'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E1:40"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="lattice5.html#D6" target="_self" title="LATTICE5:def.6">Def6</a></i>
;<br/><i><font color="Green" title="E42">A119</font></i>: <font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
<b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>
<br/>.= 
<font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
<b>by </b><i><a class="ref" href="yellow_5.html#T1" title="YELLOW_5:th.1">YELLOW_5:1</a></i>
<br/>.= 
<font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
<b>by </b><i><a class="ref" href="yellow_5.html#T1" title="YELLOW_5:th.1">YELLOW_5:1</a></i>
<br/>.= 
<font color="Maroon" title="c10">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>
;<br/><a NAME="E7:40_7_1_7"/>
<font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">u'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">u'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span>
 ;<br/><a NAME="E8:40_7_1_7"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">u'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">u'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:40"><i><font color="Green" title="E29">A3</font></i></a>, <a class="ref" href="yellow_3.html#T3" title="YELLOW_3:th.3">YELLOW_3:3</a></i>;<br/><a NAME="E9:40_7_1_7"/><b>then </b><i><font color="Green" title="E43">A120</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">u'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:40_7_1_7"><i><font color="Green" title="E39">A116</font></i></a>, <a class="txt" href="#E4:40_7_1_7"><i><font color="Green" title="E40">A117</font></i></a>, <a class="txt" href="#E5:40_7_1_7"><i><font color="Green" title="E41">A118</font></i></a>, <a class="ref" href="orders_2.html#T26" title="ORDERS_2:th.26">ORDERS_2:26</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">u'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">u'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">u'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
<b>by </b><i><a class="txt" href="#E6:40_7_1_7"><i><font color="Green" title="E42">A119</font></i></a>, <a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">u'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:40_7_1_7"><i><font color="Green" title="E38">A115</font></i></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>
;<br/><b>hence </b><a NAME="E11:40_7_1_7"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:40_7_1_7"><i><font color="Green" title="E38">A115</font></i></a>, <a class="txt" href="#E9:40_7_1_7"><i><font color="Green" title="E43">A120</font></i></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_7_1_8"><b>suppose </b></a><a NAME="E1:40_7_1_8"/><i><font color="Green" title="E38">A121</font></i>: 
( <font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><b>then </b><i><font color="Green" title="E39">A122</font></i>: <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">u'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
<b>by </b><i><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">u'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>
;<br/><a NAME="E3:40_7_1_8"/>
<font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">u'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">u'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:40"><i><font color="Green" title="E28">A2</font></i></a>, <a class="ref" href="lattice5.html#D8" target="_self" title="LATTICE5:def.8">Def8</a></i>;<br/><a NAME="E4:40_7_1_8"/><b>then </b><i><font color="Green" title="E40">A123</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">u'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">u'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
 <b>by </b><i><a class="ref" href="waybel_1.html#T3" title="WAYBEL_1:th.3">WAYBEL_1:3</a></i>;<br/><i><font color="Green" title="E41">A124</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">u'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">u'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">u'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E1:40"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="lattice5.html#D6" target="_self" title="LATTICE5:def.6">Def6</a></i>
;<br/><a NAME="E6:40_7_1_8"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">u'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">u'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span>
 ;<br/><a NAME="E7:40_7_1_8"/><b>then </b><i><font color="Green" title="E42">A125</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">u'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">u'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:40"><i><font color="Green" title="E29">A3</font></i></a>, <a class="ref" href="yellow_3.html#T3" title="YELLOW_3:th.3">YELLOW_3:3</a></i>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span> = 
<font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:40_7_1_8"><i><font color="Green" title="E38">A121</font></i></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>
<br/>.= 
<font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">u'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E1:40_7_1_8"><i><font color="Green" title="E38">A121</font></i></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">u'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">u'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="yellow_5.html#T1" title="YELLOW_5:th.1">YELLOW_5:1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">u'</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>
;<br/><b>hence </b><a NAME="E9:40_7_1_8"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:40_7_1_8"><i><font color="Green" title="E39">A122</font></i></a>, <a class="txt" href="#E4:40_7_1_8"><i><font color="Green" title="E40">A123</font></i></a>, <a class="txt" href="#E5:40_7_1_8"><i><font color="Green" title="E41">A124</font></i></a>, <a class="txt" href="#E7:40_7_1_8"><i><font color="Green" title="E42">A125</font></i></a>, <a class="ref" href="orders_2.html#T26" title="ORDERS_2:th.26">ORDERS_2:26</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_7_1_9"><b>suppose </b></a><a NAME="E1:40_7_1_9"/><i><font color="Green" title="E38">A126</font></i>: 
( <font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font>
<b>by </b><i><a class="txt" href="#E1:40_7_1_9"><i><font color="Green" title="E38">A126</font></i></a>, <a class="ref" href="waybel_1.html#T4" title="WAYBEL_1:th.4">WAYBEL_1:4</a></i>
;<br/><b>hence </b><a NAME="E3:40_7_1_9"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E13:40"/><i><font color="Green" title="E37">A127</font></i>: 
for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font>, <font color="Olive" title="b3">u</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice5.html#K6" title="LATTICE5:func.6">new_set</a> <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">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="c8">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="c8">B</font> holds <br/><span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b3">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Olive" title="b2">q</font>,<font color="Olive" title="b3">u</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">p</font>, <font color="Maroon" title="c12">q</font>, <font color="Maroon" title="c13">u</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice5.html#K6" title="LATTICE5:func.6">new_set</a> <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> implies <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:40_8"/><i><font color="Green" title="E38">A128</font></i>: 
( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_8_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:40_8_1"/>
( ( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) )
 </span><b>by </b><i><a class="txt" href="#E1:40_8"><i><font color="Green" title="E38">A128</font></i></a>, <a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_8_1_1"><b>suppose </b></a><a NAME="E1:40_8_1_1"/><i><font color="Green" title="E39">A129</font></i>: 
( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><a NAME="E2:40_8_1_1"/>
 <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c2">L</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_0.html#T44" title="YELLOW_0:th.44">YELLOW_0:44</a></i>;<br/><b>hence </b><a NAME="E3:40_8_1_1"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:40_8_1_1"><i><font color="Green" title="E39">A129</font></i></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_8_1_2"><b>suppose </b></a><a NAME="E1:40_8_1_2"/><i><font color="Green" title="E39">A130</font></i>: 
( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
<b>by </b><i><a class="txt" href="#E1:40_8_1_2"><i><font color="Green" title="E39">A130</font></i></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>
<br/>.= 
<font color="Maroon" title="c10">b</font>
<b>by </b><i><a class="ref" href="waybel_1.html#T4" title="WAYBEL_1:th.4">WAYBEL_1:4</a></i>
;<br/><b>hence </b><a NAME="E3:40_8_1_2"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:40_8_1_2"><i><font color="Green" title="E39">A130</font></i></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_8_1_3"><b>suppose </b></a><a NAME="E1:40_8_1_3"/><i><font color="Green" title="E39">A131</font></i>: 
( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:40_8_1_3"><i><font color="Green" title="E39">A131</font></i></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>
<br/>.= 
<font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
<b>by </b><i><a class="ref" href="waybel_1.html#T4" title="WAYBEL_1:th.4">WAYBEL_1:4</a></i>
;<br/><b>hence </b><a NAME="E3:40_8_1_3"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:40_8_1_3"><i><font color="Green" title="E39">A131</font></i></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_8_1_4"><b>suppose </b></a><a NAME="E1:40_8_1_4"/><i><font color="Green" title="E39">A132</font></i>: 
( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><a NAME="E2:40_8_1_4"/>
 <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c2">L</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_0.html#T44" title="YELLOW_0:th.44">YELLOW_0:44</a></i>;<br/><b>hence </b><a NAME="E3:40_8_1_4"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:40_8_1_4"><i><font color="Green" title="E39">A132</font></i></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_8_1_5"><b>suppose </b></a><a NAME="E1:40_8_1_5"/><i><font color="Green" title="E39">A133</font></i>: 
( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span> = 
<font color="Maroon" title="c10">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
<b>by </b><i><a class="txt" href="#E1:40_8_1_5"><i><font color="Green" title="E39">A133</font></i></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>
<br/>.= 
<font color="Maroon" title="c10">b</font>
<b>by </b><i><a class="ref" href="waybel_1.html#T4" title="WAYBEL_1:th.4">WAYBEL_1:4</a></i>
;<br/><b>hence </b><a NAME="E3:40_8_1_5"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:40_8_1_5"><i><font color="Green" title="E39">A133</font></i></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_8_1_6"><b>suppose </b></a><a NAME="E1:40_8_1_6"/><i><font color="Green" title="E39">A134</font></i>: 
( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span> = 
<font color="Maroon" title="c10">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>
<br/>.= 
<font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
<b>by </b><i><a class="txt" href="#E1:40_8_1_6"><i><font color="Green" title="E39">A134</font></i></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>
;<br/><b>hence </b><a NAME="E3:40_8_1_6"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:40_8_1_6"><i><font color="Green" title="E39">A134</font></i></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_8_1_7"><b>suppose </b></a><a NAME="E1:40_8_1_7"/><i><font color="Green" title="E39">A135</font></i>: 
( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><a NAME="E2:40_8_1_7"/>
 <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c2">L</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_0.html#T44" title="YELLOW_0:th.44">YELLOW_0:44</a></i>;<br/><b>hence </b><a NAME="E3:40_8_1_7"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:40_8_1_7"><i><font color="Green" title="E39">A135</font></i></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_8_1_8"><b>suppose </b></a><a NAME="E1:40_8_1_8"/><i><font color="Green" title="E39">A136</font></i>: 
( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><a NAME="E2:40_8_1_8"/><b>then </b><i><font color="Green" title="E40">A137</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b</font>
 <b>by </b><i><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:40_8_1_8"><i><font color="Green" title="E39">A136</font></i></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font>
<b>by </b><i><a class="txt" href="#E1:40_8_1_8"><i><font color="Green" title="E39">A136</font></i></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>
<br/>.= 
<font color="Maroon" title="c10">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>
<br/>.= 
<font color="Maroon" title="c10">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font>
<b>by </b><i><a class="ref" href="yellow_5.html#T1" title="YELLOW_5:th.1">YELLOW_5:1</a></i>
;<br/><b>hence </b><a NAME="E4:40_8_1_8"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:40_8_1_8"><i><font color="Green" title="E40">A137</font></i></a>, <a class="ref" href="yellow_0.html#T22" title="YELLOW_0:th.22">YELLOW_0:22</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_8_1_9"><b>suppose </b></a><a NAME="E1:40_8_1_9"/><i><font color="Green" title="E39">A138</font></i>: 
( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:40_8_1_9"><i><font color="Green" title="E39">A138</font></i></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>
<br/>.= 
<font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
<b>by </b><i><a class="ref" href="waybel_1.html#T4" title="WAYBEL_1:th.4">WAYBEL_1:4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font>
<b>by </b><i><a class="txt" href="#E1:40_8_1_9"><i><font color="Green" title="E39">A138</font></i></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>
;<br/><b>hence </b><a NAME="E3:40_8_1_9"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:40_8_1_9"><i><font color="Green" title="E39">A138</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_8_1_10"><b>suppose </b></a><a NAME="E1:40_8_1_10"/><i><font color="Green" title="E39">A139</font></i>: 
( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span> = 
<font color="Maroon" title="c10">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
<b>by </b><i><a class="txt" href="#E1:40_8_1_10"><i><font color="Green" title="E39">A139</font></i></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>
<br/>.= 
<font color="Maroon" title="c10">b</font>
<b>by </b><i><a class="ref" href="waybel_1.html#T4" title="WAYBEL_1:th.4">WAYBEL_1:4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font>
<b>by </b><i><a class="txt" href="#E1:40_8_1_10"><i><font color="Green" title="E39">A139</font></i></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>
;<br/><b>hence </b><a NAME="E3:40_8_1_10"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:40_8_1_10"><i><font color="Green" title="E39">A139</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_8_1_11"><b>suppose </b></a><a NAME="E1:40_8_1_11"/><i><font color="Green" title="E39">A140</font></i>: 
( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><a NAME="E2:40_8_1_11"/>
 <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c2">L</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_0.html#T44" title="YELLOW_0:th.44">YELLOW_0:44</a></i>;<br/><b>hence </b><a NAME="E3:40_8_1_11"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:40_8_1_11"><i><font color="Green" title="E39">A140</font></i></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_8_1_12"><b>suppose </b></a><a NAME="E1:40_8_1_12"/><i><font color="Green" title="E39">A141</font></i>: 
( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><a NAME="E2:40_8_1_12"/><b>then </b><i><font color="Green" title="E40">A142</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a</font>
 <b>by </b><i><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span> = 
<font color="Maroon" title="c10">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:40_8_1_12"><i><font color="Green" title="E39">A141</font></i></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>
<br/>.= 
<font color="Maroon" title="c10">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:40_8_1_12"><i><font color="Green" title="E39">A141</font></i></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font>
<b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>
<br/>.= 
<font color="Maroon" title="c10">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font>
<b>by </b><i><a class="ref" href="yellow_5.html#T1" title="YELLOW_5:th.1">YELLOW_5:1</a></i>
;<br/><b>hence </b><a NAME="E4:40_8_1_12"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:40_8_1_12"><i><font color="Green" title="E40">A142</font></i></a>, <a class="ref" href="yellow_0.html#T22" title="YELLOW_0:th.22">YELLOW_0:22</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_8_1_13"><b>suppose </b></a><a NAME="E1:40_8_1_13"/><i><font color="Green" title="E39">A143</font></i>: 
( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
<b>by </b><i><a class="txt" href="#E1:40_8_1_13"><i><font color="Green" title="E39">A143</font></i></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>
<br/>.= 
<font color="Maroon" title="c10">b</font>
<b>by </b><i><a class="ref" href="waybel_1.html#T4" title="WAYBEL_1:th.4">WAYBEL_1:4</a></i>
;<br/><b>hence </b><a NAME="E3:40_8_1_13"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:40_8_1_13"><i><font color="Green" title="E39">A143</font></i></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_8_1_14"><b>suppose </b></a><a NAME="E1:40_8_1_14"/><i><font color="Green" title="E39">A144</font></i>: 
( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><a NAME="E2:40_8_1_14"/>
 <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c2">L</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_0.html#T44" title="YELLOW_0:th.44">YELLOW_0:44</a></i>;<br/><b>hence </b><a NAME="E3:40_8_1_14"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:40_8_1_14"><i><font color="Green" title="E39">A144</font></i></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_8_1_15"><b>suppose </b></a><a NAME="E1:40_8_1_15"/><i><font color="Green" title="E39">A145</font></i>: 
( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font>
<b>by </b><i><a class="txt" href="#E1:40_8_1_15"><i><font color="Green" title="E39">A145</font></i></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>
<br/>.= 
<font color="Maroon" title="c9">a</font>
<b>by </b><i><a class="ref" href="waybel_1.html#T4" title="WAYBEL_1:th.4">WAYBEL_1:4</a></i>
;<br/><b>hence </b><a NAME="E3:40_8_1_15"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:40_8_1_15"><i><font color="Green" title="E39">A145</font></i></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_8_1_16"><b>suppose </b></a><a NAME="E1:40_8_1_16"/><i><font color="Green" title="E39">A146</font></i>: 
( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><a NAME="E2:40_8_1_16"/><b>then </b><i><font color="Green" title="E40">A147</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b</font>
 <b>by </b><i><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span> = 
<font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:40_8_1_16"><i><font color="Green" title="E39">A146</font></i></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>
<br/>.= 
<font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:40_8_1_16"><i><font color="Green" title="E39">A146</font></i></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
<b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>
<br/>.= 
<font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
<b>by </b><i><a class="ref" href="yellow_5.html#T1" title="YELLOW_5:th.1">YELLOW_5:1</a></i>
;<br/><b>hence </b><a NAME="E4:40_8_1_16"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:40_8_1_16"><i><font color="Green" title="E40">A147</font></i></a>, <a class="ref" href="yellow_0.html#T22" title="YELLOW_0:th.22">YELLOW_0:22</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_8_1_17"><b>suppose </b></a><a NAME="E1:40_8_1_17"/><i><font color="Green" title="E39">A148</font></i>: 
( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><a NAME="E2:40_8_1_17"/>
 <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c2">L</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_0.html#T44" title="YELLOW_0:th.44">YELLOW_0:44</a></i>;<br/><b>hence </b><a NAME="E3:40_8_1_17"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:40_8_1_17"><i><font color="Green" title="E39">A148</font></i></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_8_1_18"><b>suppose </b></a><a NAME="E1:40_8_1_18"/><i><font color="Green" title="E39">A149</font></i>: 
( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span> = 
<font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font>
<b>by </b><i><a class="txt" href="#E1:40_8_1_18"><i><font color="Green" title="E39">A149</font></i></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>
<br/>.= 
<font color="Maroon" title="c9">a</font>
<b>by </b><i><a class="ref" href="waybel_1.html#T4" title="WAYBEL_1:th.4">WAYBEL_1:4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font>
<b>by </b><i><a class="txt" href="#E1:40_8_1_18"><i><font color="Green" title="E39">A149</font></i></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>
;<br/><b>hence </b><a NAME="E3:40_8_1_18"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:40_8_1_18"><i><font color="Green" title="E39">A149</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_8_1_19"><b>suppose </b></a><a NAME="E1:40_8_1_19"/><i><font color="Green" title="E39">A150</font></i>: 
( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:40_8_1_19"><i><font color="Green" title="E39">A150</font></i></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>
<br/>.= 
<font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
<b>by </b><i><a class="ref" href="waybel_1.html#T4" title="WAYBEL_1:th.4">WAYBEL_1:4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font>
<b>by </b><i><a class="txt" href="#E1:40_8_1_19"><i><font color="Green" title="E39">A150</font></i></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>
;<br/><b>hence </b><a NAME="E3:40_8_1_19"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:40_8_1_19"><i><font color="Green" title="E39">A150</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_8_1_20"><b>suppose </b></a><a NAME="E1:40_8_1_20"/><i><font color="Green" title="E39">A151</font></i>: 
( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><a NAME="E2:40_8_1_20"/><b>then </b><i><font color="Green" title="E40">A152</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a</font>
 <b>by </b><i><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:40_8_1_20"><i><font color="Green" title="E39">A151</font></i></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
<b>by </b><i><a class="txt" href="#E1:40_8_1_20"><i><font color="Green" title="E39">A151</font></i></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>
<br/>.= 
<font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>
<br/>.= 
<font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
<b>by </b><i><a class="ref" href="yellow_5.html#T1" title="YELLOW_5:th.1">YELLOW_5:1</a></i>
;<br/><b>hence </b><a NAME="E4:40_8_1_20"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:40_8_1_20"><i><font color="Green" title="E40">A152</font></i></a>, <a class="ref" href="yellow_0.html#T22" title="YELLOW_0:th.22">YELLOW_0:22</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_8_1_21"><b>suppose </b></a><a NAME="E1:40_8_1_21"/><i><font color="Green" title="E39">A153</font></i>: 
( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><a NAME="E2:40_8_1_21"/>
 <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c2">L</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_0.html#T44" title="YELLOW_0:th.44">YELLOW_0:44</a></i>;<br/><b>hence </b><a NAME="E3:40_8_1_21"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:40_8_1_21"><i><font color="Green" title="E39">A153</font></i></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_8_1_22"><b>suppose </b></a><a NAME="E1:40_8_1_22"/><i><font color="Green" title="E39">A154</font></i>: 
( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span> = 
<font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>
<br/>.= 
<font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
<b>by </b><i><a class="txt" href="#E1:40_8_1_22"><i><font color="Green" title="E39">A154</font></i></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>
;<br/><b>hence </b><a NAME="E3:40_8_1_22"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:40_8_1_22"><i><font color="Green" title="E39">A154</font></i></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_8_1_23"><b>suppose </b></a><a NAME="E1:40_8_1_23"/><i><font color="Green" title="E39">A155</font></i>: 
( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span> = 
<font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font>
<b>by </b><i><a class="txt" href="#E1:40_8_1_23"><i><font color="Green" title="E39">A155</font></i></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>
<br/>.= 
<font color="Maroon" title="c9">a</font>
<b>by </b><i><a class="ref" href="waybel_1.html#T4" title="WAYBEL_1:th.4">WAYBEL_1:4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font>
<b>by </b><i><a class="txt" href="#E1:40_8_1_23"><i><font color="Green" title="E39">A155</font></i></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>
;<br/><b>hence </b><a NAME="E3:40_8_1_23"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:40_8_1_23"><i><font color="Green" title="E39">A155</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_8_1_24"><b>suppose </b></a><a NAME="E1:40_8_1_24"/><i><font color="Green" title="E39">A156</font></i>: 
( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><a NAME="E2:40_8_1_24"/>
 <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c2">L</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_0.html#T44" title="YELLOW_0:th.44">YELLOW_0:44</a></i>;<br/><b>hence </b><a NAME="E3:40_8_1_24"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:40_8_1_24"><i><font color="Green" title="E39">A156</font></i></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_8_1_25"><b>suppose </b></a><a NAME="E1:40_8_1_25"/><i><font color="Green" title="E39">A157</font></i>: 
( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:40_8_1_25"><i><font color="Green" title="E39">A157</font></i></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>
<br/>.= 
<font color="Maroon" title="c9">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">b</font>
<b>by </b><i><a class="ref" href="waybel_1.html#T4" title="WAYBEL_1:th.4">WAYBEL_1:4</a></i>
;<br/><b>hence </b><a NAME="E3:40_8_1_25"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:40_8_1_25"><i><font color="Green" title="E39">A157</font></i></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_8_1_26"><b>suppose </b></a><a NAME="E1:40_8_1_26"/><i><font color="Green" title="E39">A158</font></i>: 
( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">a</font>
<b>by </b><i><a class="txt" href="#E1:40_8_1_26"><i><font color="Green" title="E39">A158</font></i></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>
<br/>.= 
<font color="Maroon" title="c9">a</font>
<b>by </b><i><a class="ref" href="waybel_1.html#T4" title="WAYBEL_1:th.4">WAYBEL_1:4</a></i>
;<br/><b>hence </b><a NAME="E3:40_8_1_26"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:40_8_1_26"><i><font color="Green" title="E39">A158</font></i></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_8_1_27"><b>suppose </b></a><a NAME="E1:40_8_1_27"/><i><font color="Green" title="E39">A159</font></i>: 
( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><a NAME="E2:40_8_1_27"/>
 <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c2">L</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_0.html#T44" title="YELLOW_0:th.44">YELLOW_0:44</a></i>;<br/><b>hence </b><a NAME="E3:40_8_1_27"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:40_8_1_27"><i><font color="Green" title="E39">A159</font></i></a>, <a class="ref" href="lattice5.html#D11" target="_self" title="LATTICE5:def.11">Def11</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E14:40"/>
for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font>, <font color="Olive" title="b3">u</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice5.html#K6" title="LATTICE5:func.6">new_set</a> <font color="Maroon" title="c1">A</font> holds  <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b3">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Olive" title="b2">q</font>,<font color="Olive" title="b3">u</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_9"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">p</font>, <font color="Maroon" title="c12">q</font>, <font color="Maroon" title="c13">u</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice5.html#K6" title="LATTICE5:func.6">new_set</a> <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_9_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:40_9_1"/>
( ( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> ) or ( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> ) or ( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> ) or ( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> ) or ( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> ) or ( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> ) or ( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> ) or ( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> ) )
 </span><b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_9_1_1"><b>suppose </b></a><a NAME="E1:40_9_1_1"/>
( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><b>hence </b><a NAME="E2:40_9_1_1"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E6:40"><i><font color="Green" title="E30">A4</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_9_1_2"><b>suppose </b></a><a NAME="E1:40_9_1_2"/>
( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><b>hence </b><a NAME="E2:40_9_1_2"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E7:40"><i><font color="Green" title="E31">A7</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_9_1_3"><b>suppose </b></a><a NAME="E1:40_9_1_3"/>
( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><b>hence </b><a NAME="E2:40_9_1_3"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E8:40"><i><font color="Green" title="E32">A21</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_9_1_4"><b>suppose </b></a><a NAME="E1:40_9_1_4"/>
( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><b>hence </b><a NAME="E2:40_9_1_4"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E9:40"><i><font color="Green" title="E33">A38</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_9_1_5"><b>suppose </b></a><a NAME="E1:40_9_1_5"/>
( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><b>hence </b><a NAME="E2:40_9_1_5"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E10:40"><i><font color="Green" title="E34">A73</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_9_1_6"><b>suppose </b></a><a NAME="E1:40_9_1_6"/>
( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><b>hence </b><a NAME="E2:40_9_1_6"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E11:40"><i><font color="Green" title="E35">A84</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_9_1_7"><b>suppose </b></a><a NAME="E1:40_9_1_7"/>
( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><b>hence </b><a NAME="E2:40_9_1_7"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E12:40"><i><font color="Green" title="E36">A98</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_9_1_8"><b>suppose </b></a><a NAME="E1:40_9_1_8"/>
( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span></span><br/><div class="add"><b>hence </b><a NAME="E2:40_9_1_8"/>
<span class="p1">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c13">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E13:40"><i><font color="Green" title="E37">A127</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E15:40"/>
 <a href="lattice5.html#K7" title="LATTICE5:func.7">new_bi_fun</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">q</font> is <a href="lattice5.html#V3" title="LATTICE5:attr.3">u.t.i.</a>
 <b>by </b><i><a class="ref" href="lattice5.html#D8" target="_self" title="LATTICE5:def.8">Def8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
