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

<span class="kw">let </span><font color="Maroon" title="c1">X</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="yellow_1.html#K2" title="YELLOW_1:func.2">BoolePoset</a> <font color="Maroon" title="c1">X</font> is  <a href="waybel_8.html#V3" title="WAYBEL_8:attr.3">arithmetic</a> </span><br/>

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="7_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">BoolePoset</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> holds   <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>, <a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">BoolePoset</a> <font color="Maroon" title="c1">X</font></span>)</span></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">BoolePoset</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>, <a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">BoolePoset</a> <font color="Maroon" title="c1">X</font></span>)</span></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c4">x9</font> = <font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c5">y9</font> = <font color="Maroon" title="c3">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">BoolePoset</a> <font color="Maroon" title="c1">X</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T58" onmouseover="rs('yellow_0/T58')" onmouseout="rh()">YELLOW_0:58</a></span>;<br/><a NAME="E2:7_1"/>
<font color="Maroon" title="c4">x9</font> is  <a href="waybel_3.html#V1" title="WAYBEL_3:attr.1">compact</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_8.html#D1" onmouseover="rs('waybel_8/D1')" onmouseout="rh()">WAYBEL_8:def 1</a></span>;<br/><a NAME="E3:7_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">x9</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_8.html#T28" onmouseover="rs('waybel_8/T28')" onmouseout="rh()">WAYBEL_8:28</a></span>;<br/><a NAME="E4:7_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">x9</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c5">y9</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 ;<br/><a NAME="E5:7_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">x9</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c5">y9</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_1.html#T17" onmouseover="rs('yellow_1/T17')" onmouseout="rh()">YELLOW_1:17</a></span>;<br/><a NAME="E6:7_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">x9</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c5">y9</font> is  <a href="waybel_3.html#V1" title="WAYBEL_3:attr.1">compact</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_8.html#T28" onmouseover="rs('waybel_8/T28')" onmouseout="rh()">WAYBEL_8:28</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">xy</font> = <font color="Maroon" title="c4">x9</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c5">y9</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">BoolePoset</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_8.html#D1" onmouseover="rs('waybel_8/D1')" onmouseout="rh()">WAYBEL_8:def 1</a></span>;<br/><div><span class="lab"><font color="Green" title="E7">A1</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="7_1_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">BoolePoset</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span>  st <font color="Olive" title="b1">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Olive" title="b1">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">y</font> holds <br/><font color="Maroon" title="c6">xy</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Olive" title="b1">z</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">z</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">BoolePoset</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Maroon" title="c7">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">y</font> implies <font color="Maroon" title="c6">xy</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c7">z</font> )</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c8">z9</font> = <font color="Maroon" title="c7">z</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">BoolePoset</a> <font color="Maroon" title="c1">X</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T58" onmouseover="rs('yellow_0/T58')" onmouseout="rh()">YELLOW_0:58</a></span>;<br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:7_1_1"/><span class="lab"><font color="Green" title="E8">A2</font></span>: 
<font color="Maroon" title="c7">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">x</font>
 <span class="kw">and </span><br/><a NAME="E3:7_1_1"/><span class="lab"><font color="Green" title="E9">A3</font></span>: 
<font color="Maroon" title="c7">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">y</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">xy</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c7">z</font></span><br/><a NAME="E4:7_1_1"/>
<font color="Maroon" title="c8">z9</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">y9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:7_1_1"><span class="lab"><font color="Green" title="E9">A3</font></span></a>, <a class="ref" href="yellow_0.html#T59" onmouseover="rs('yellow_0/T59')" onmouseout="rh()">YELLOW_0:59</a></span>;<br/><a NAME="E5:7_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E10">A4</font></span>: 
<font color="Maroon" title="c8">z9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">y9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_1.html#T2" onmouseover="rs('yellow_1/T2')" onmouseout="rh()">YELLOW_1:2</a></span>;<br/><a NAME="E6:7_1_1"/>
<font color="Maroon" title="c8">z9</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">x9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:7_1_1"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="ref" href="yellow_0.html#T59" onmouseover="rs('yellow_0/T59')" onmouseout="rh()">YELLOW_0:59</a></span>;<br/><a NAME="E7:7_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">z9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">x9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_1.html#T2" onmouseover="rs('yellow_1/T2')" onmouseout="rh()">YELLOW_1:2</a></span>;<br/><a NAME="E8:7_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">z9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">x9</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c5">y9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:7_1_1"><span class="lab"><font color="Green" title="E10">A4</font></span></a>, <a class="ref" href="xboole_1.html#T19" onmouseover="rs('xboole_1/T19')" onmouseout="rh()">XBOOLE_1:19</a></span>;<br/><a NAME="E9:7_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">z9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">x9</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c5">y9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_1.html#T17" onmouseover="rs('yellow_1/T17')" onmouseout="rh()">YELLOW_1:17</a></span>;<br/><a NAME="E10:7_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">z9</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">x9</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c5">y9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_1.html#T2" onmouseover="rs('yellow_1/T2')" onmouseout="rh()">YELLOW_1:2</a></span>;<br/><span class="kw">hence </span><a NAME="E11:7_1_1"/>
<font color="Maroon" title="c6">xy</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c7">z</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T60" onmouseover="rs('yellow_0/T60')" onmouseout="rh()">YELLOW_0:60</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E9:7_1"/>
<font color="Maroon" title="c4">x9</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c5">y9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">y9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T17" onmouseover="rs('xboole_1/T17')" onmouseout="rh()">XBOOLE_1:17</a></span>;<br/><a NAME="E10:7_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">x9</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c5">y9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">y9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_1.html#T17" onmouseover="rs('yellow_1/T17')" onmouseout="rh()">YELLOW_1:17</a></span>;<br/><a NAME="E11:7_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">x9</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c5">y9</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">y9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_1.html#T2" onmouseover="rs('yellow_1/T2')" onmouseout="rh()">YELLOW_1:2</a></span>;<br/><a NAME="E12:7_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E8">A5</font></span>: 
<font color="Maroon" title="c6">xy</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T60" onmouseover="rs('yellow_0/T60')" onmouseout="rh()">YELLOW_0:60</a></span>;<br/><a NAME="E13:7_1"/>
<font color="Maroon" title="c4">x9</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c5">y9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">x9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T17" onmouseover="rs('xboole_1/T17')" onmouseout="rh()">XBOOLE_1:17</a></span>;<br/><a NAME="E14:7_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">x9</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c5">y9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">x9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_1.html#T17" onmouseover="rs('yellow_1/T17')" onmouseout="rh()">YELLOW_1:17</a></span>;<br/><a NAME="E15:7_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">x9</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c5">y9</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">x9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_1.html#T2" onmouseover="rs('yellow_1/T2')" onmouseout="rh()">YELLOW_1:2</a></span>;<br/><a NAME="E16:7_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">xy</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T60" onmouseover="rs('yellow_0/T60')" onmouseout="rh()">YELLOW_0:60</a></span>;<br/><span class="kw">hence </span><a NAME="E17:7_1"/>
 <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>, <a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">BoolePoset</a> <font color="Maroon" title="c1">X</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:7_1"><span class="lab"><font color="Green" title="E8">A5</font></span></a>, <a class="txt" href="#E8:7_1"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="ref" href="yellow_0.html#T19" onmouseover="rs('yellow_0/T19')" onmouseout="rh()">YELLOW_0:19</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E2:7"/><span class="kw">then </span>
 <a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">BoolePoset</a> <font color="Maroon" title="c1">X</font></span>)</span> is  <a href="lattice3.html#V2" title="LATTICE3:attr.2">with_infima</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T21" onmouseover="rs('yellow_0/T21')" onmouseout="rh()">YELLOW_0:21</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:7"/>
 <a href="yellow_1.html#K2" title="YELLOW_1:func.2">BoolePoset</a> <font color="Maroon" title="c1">X</font> is  <a href="waybel_8.html#V3" title="WAYBEL_8:attr.3">arithmetic</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_8.html#T19" onmouseover="rs('waybel_8/T19')" onmouseout="rh()">WAYBEL_8:19</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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