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<span class="kw">let </span><font color="Maroon" title="c1">L</font> be   <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a>   <a href="waybel_1.html#V2" title="WAYBEL_1:attr.2">distributive</a>  <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st ( for <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b2">b</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Olive" title="b1">a</font> holds <br/> <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="lattice7.html#K3" title="LATTICE7:func.3">Join-IRR</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> ) holds <br/> <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="lattice7.html#K3" title="LATTICE7:func.3">Join-IRR</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( for <font color="Olive" title="b1">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b1">b</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c2">a</font> holds <br/> <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Olive" title="b1">b</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="lattice7.html#K3" title="LATTICE7:func.3">Join-IRR</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> ) implies  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="lattice7.html#K3" title="LATTICE7:func.3">Join-IRR</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> )</span><br/>



<span class="kw">assume </span><a NAME="E1:21"/><span class="lab"><font color="Green" title="E16">A1</font></span>: 
for <font color="Olive" title="b1">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b1">b</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c2">a</font> holds <br/> <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Olive" title="b1">b</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="lattice7.html#K3" title="LATTICE7:func.3">Join-IRR</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="lattice7.html#K3" title="LATTICE7:func.3">Join-IRR</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font></span><br/>

<span class="kw">thus </span><a NAME="E2:21"/>
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="lattice7.html#K3" title="LATTICE7:func.3">Join-IRR</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:21_1_1"/>
( <font color="Maroon" title="c2">a</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="c1">L</font> or ( not <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lattice7.html#K3" title="LATTICE7:func.3">Join-IRR</a> <font color="Maroon" title="c1">L</font> &amp; <font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font> ) or <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lattice7.html#K3" title="LATTICE7:func.3">Join-IRR</a> <font color="Maroon" title="c1">L</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:21_1_1_1"/><span class="lab"><font color="Green" title="E17">A2</font></span>: 
<font color="Maroon" title="c2">a</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="c1">L</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="lattice7.html#K3" title="LATTICE7:func.3">Join-IRR</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font></span><br/><div class="add"><a NAME="E2:21_1_1_1"/><span class="lab"><font color="Green" title="E18">A3</font></span>: 
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="lattice7.html#K3" title="LATTICE7:func.3">Join-IRR</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">consider </span><font color="Maroon" title="c3">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="lattice7.html#K3" title="LATTICE7:func.3">Join-IRR</a> <font color="Maroon" title="c1">L</font></span>)</span>;<br/>
<span class="kw">assume </span><a NAME="E2:21_1_1_1_1"/><span class="lab"><font color="Green" title="E19">A4</font></span>: 
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="lattice7.html#K3" title="LATTICE7:func.3">Join-IRR</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E3:21_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E4:21_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A5</font></span>: 
<font color="Maroon" title="c3">x</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="c1">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E5:21_1_1_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lattice7.html#K3" title="LATTICE7:func.3">Join-IRR</a> <font color="Maroon" title="c1">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21_1_1_1_1"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:21_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:21_1_1_1_1"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="ref" href="lattice7.html#T10" target="_self" title="LATTICE7:th.10">Th10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E3:21_1_1_1"/>
 <a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_1_1_1"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="ref" href="waybel_4.html#T23" title="WAYBEL_4:th.23">WAYBEL_4:23</a></span>;<br/><span class="kw">hence </span><a NAME="E4:21_1_1_1"/>
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="lattice7.html#K3" title="LATTICE7:func.3">Join-IRR</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_1_1_1"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="txt" href="#E2:21_1_1_1"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="ref" href="yellow_0.html#D11" title="YELLOW_0:def.11">YELLOW_0:def 11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:21_1_1_2"/>
( not <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lattice7.html#K3" title="LATTICE7:func.3">Join-IRR</a> <font color="Maroon" title="c1">L</font> &amp; <font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="lattice7.html#K3" title="LATTICE7:func.3">Join-IRR</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">y</font>, <font color="Maroon" title="c4">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><span class="kw"> such that </span><br/><a NAME="E3:21_1_1_2"/><span class="lab"><font color="Green" title="E17">A6</font></span>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c4">z</font>
 <span class="kw">and </span><br/><a NAME="E4:21_1_1_2"/><span class="lab"><font color="Green" title="E18">A7</font></span>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">y</font>
 <span class="kw">and </span><br/><a NAME="E5:21_1_1_2"/><span class="lab"><font color="Green" title="E19">A8</font></span>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">z</font>
 ;<br/><a NAME="E6:21_1_1_2"/><span class="lab"><font color="Green" title="E20">A9</font></span>: 
<font color="Maroon" title="c3">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21_1_1_2"><span class="lab"><font color="Green" title="E17">A6</font></span></a>, <a class="ref" href="yellow_0.html#T22" title="YELLOW_0:th.22">YELLOW_0:22</a></span>;<br/><a NAME="E7:21_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A10</font></span>: 
<font color="Maroon" title="c3">y</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c2">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:21_1_1_2"><span class="lab"><font color="Green" title="E18">A7</font></span></a>, <a class="ref" href="orders_2.html#D10" title="ORDERS_2:def.10">ORDERS_2:def 10</a></span>;<br/><a NAME="E8:21_1_1_2"/><span class="lab"><font color="Green" title="E22">A11</font></span>: 
<span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="lattice7.html#K3" title="LATTICE7:func.3">Join-IRR</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="lattice7.html#R1" title="LATTICE7:pred.1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="lattice7.html#K3" title="LATTICE7:func.3">Join-IRR</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="lattice7.html#K3" title="LATTICE7:func.3">Join-IRR</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="lattice7.html#D1" target="_self" title="LATTICE7:def.1">LATTICE7:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="lattice7.html#K3" title="LATTICE7:func.3">Join-IRR</a> <font color="Maroon" title="c1">L</font></span>)</span> implies <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="lattice7.html#K3" title="LATTICE7:func.3">Join-IRR</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="lattice7.html#K3" title="LATTICE7:func.3">Join-IRR</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c6">x1</font> = <font color="Maroon" title="c5">x</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">y1</font> = <font color="Maroon" title="c3">y</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">a1</font> = <font color="Maroon" title="c2">a</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">z1</font> = <font color="Maroon" title="c4">z</font>;<br/>
<span class="kw">assume </span><a NAME="E1:21_1_1_2_1"/><span class="lab"><font color="Green" title="E23">A12</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="lattice7.html#K3" title="LATTICE7:func.3">Join-IRR</a> <font color="Maroon" title="c1">L</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="lattice7.html#K3" title="LATTICE7:func.3">Join-IRR</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="lattice7.html#K3" title="LATTICE7:func.3">Join-IRR</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span></span><br/>

<a NAME="E2:21_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A13</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lattice7.html#K3" title="LATTICE7:func.3">Join-IRR</a> <font color="Maroon" title="c1">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E3:21_1_1_2_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c2">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_1_1_2_1"><span class="lab"><font color="Green" title="E23">A12</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E4:21_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A14</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_0.html#T17" title="WAYBEL_0:th.17">WAYBEL_0:17</a></span>;<br/>
<a NAME="E5:21_1_1_2_1"/>
<font color="Maroon" title="c5">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c4">z</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21_1_1_2"><span class="lab"><font color="Green" title="E17">A6</font></span></a>, <a class="ref" href="waybel_1.html#D3" title="WAYBEL_1:def.3">WAYBEL_1:def 3</a></span>;<br/>
<a NAME="E6:21_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c4">z</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:21_1_1_2_1"><span class="lab"><font color="Green" title="E25">A14</font></span></a>, <a class="ref" href="yellow_0.html#T25" title="YELLOW_0:th.25">YELLOW_0:25</a></span>;<br/>
<a NAME="E7:21_1_1_2_1"/><span class="kw">then </span>
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c3">y</font> or <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c4">z</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21_1_1_2_1"><span class="lab"><font color="Green" title="E24">A13</font></span></a>, <a class="ref" href="lattice7.html#T10" target="_self" title="LATTICE7:th.10">Th10</a></span>;<br/>
<a NAME="E8:21_1_1_2_1"/><span class="kw">then </span>
( <font color="Maroon" title="c5">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">y</font> or <font color="Maroon" title="c5">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">z</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T25" title="YELLOW_0:th.25">YELLOW_0:25</a></span>;<br/>
<a NAME="E9:21_1_1_2_1"/><span class="kw">then </span>
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c3">y</font> or <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c4">z</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_0.html#T17" title="WAYBEL_0:th.17">WAYBEL_0:17</a></span>;<br/>
<a NAME="E10:21_1_1_2_1"/><span class="kw">then </span>
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="lattice7.html#K3" title="LATTICE7:func.3">Join-IRR</a> <font color="Maroon" title="c1">L</font></span>)</span> or <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="lattice7.html#K3" title="LATTICE7:func.3">Join-IRR</a> <font color="Maroon" title="c1">L</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21_1_1_2_1"><span class="lab"><font color="Green" title="E24">A13</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:21_1_1_2_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="lattice7.html#K3" title="LATTICE7:func.3">Join-IRR</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="lattice7.html#K3" title="LATTICE7:func.3">Join-IRR</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E9:21_1_1_2"/><span class="lab"><font color="Green" title="E23">A15</font></span>: 
(  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="lattice7.html#K3" title="LATTICE7:func.3">Join-IRR</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="lattice7.html#K3" title="LATTICE7:func.3">Join-IRR</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">L</font> &amp;  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="lattice7.html#K3" title="LATTICE7:func.3">Join-IRR</a> <font color="Maroon" title="c1">L</font></span>)</span>,<font color="Maroon" title="c1">L</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T17" title="YELLOW_0:th.17">YELLOW_0:17</a></span>;<br/><a NAME="E10:21_1_1_2"/><span class="lab"><font color="Green" title="E24">A16</font></span>: 
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="lattice7.html#K3" title="LATTICE7:func.3">Join-IRR</a> <font color="Maroon" title="c1">L</font></span>)</span>,<font color="Maroon" title="c1">L</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T17" title="YELLOW_0:th.17">YELLOW_0:17</a></span>;<br/><a NAME="E11:21_1_1_2"/><span class="lab"><font color="Green" title="E25">A17</font></span>: 
<font color="Maroon" title="c4">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21_1_1_2"><span class="lab"><font color="Green" title="E17">A6</font></span></a>, <a class="ref" href="yellow_0.html#T22" title="YELLOW_0:th.22">YELLOW_0:22</a></span>;<br/><a NAME="E12:21_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="lattice7.html#K3" title="LATTICE7:func.3">Join-IRR</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="lattice7.html#K3" title="LATTICE7:func.3">Join-IRR</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> <a href="lattice7.html#R1" title="LATTICE7:pred.1">c=</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="lattice7.html#K3" title="LATTICE7:func.3">Join-IRR</a> <font color="Maroon" title="c1">L</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="lattice7.html#D1" target="_self" title="LATTICE7:def.1">LATTICE7:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="lattice7.html#K3" title="LATTICE7:func.3">Join-IRR</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="lattice7.html#K3" title="LATTICE7:func.3">Join-IRR</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> implies <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="lattice7.html#K3" title="LATTICE7:func.3">Join-IRR</a> <font color="Maroon" title="c1">L</font></span>)</span> )</span><br/>

<a NAME="E1:21_1_1_2_2"/>
( <span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="lattice7.html#K3" title="LATTICE7:func.3">Join-IRR</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="lattice7.html#R1" title="LATTICE7:pred.1">c=</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="lattice7.html#K3" title="LATTICE7:func.3">Join-IRR</a> <font color="Maroon" title="c1">L</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="lattice7.html#K3" title="LATTICE7:func.3">Join-IRR</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="lattice7.html#R1" title="LATTICE7:pred.1">c=</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="lattice7.html#K3" title="LATTICE7:func.3">Join-IRR</a> <font color="Maroon" title="c1">L</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:21_1_1_2"><span class="lab"><font color="Green" title="E20">A9</font></span></a>, <a class="txt" href="#E11:21_1_1_2"><span class="lab"><font color="Green" title="E25">A17</font></span></a>, <a class="ref" href="waybel_0.html#T21" title="WAYBEL_0:th.21">WAYBEL_0:21</a>, <a class="ref" href="xboole_1.html#T26" title="XBOOLE_1:th.26">XBOOLE_1:26</a></span>;<br/>
<a NAME="E2:21_1_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A18</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="lattice7.html#K3" title="LATTICE7:func.3">Join-IRR</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="lattice7.html#K3" title="LATTICE7:func.3">Join-IRR</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> <a href="lattice7.html#R1" title="LATTICE7:pred.1">c=</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="lattice7.html#K3" title="LATTICE7:func.3">Join-IRR</a> <font color="Maroon" title="c1">L</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:21_1_1_2_2"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="lattice7.html#K3" title="LATTICE7:func.3">Join-IRR</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="lattice7.html#K3" title="LATTICE7:func.3">Join-IRR</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="lattice7.html#K3" title="LATTICE7:func.3">Join-IRR</a> <font color="Maroon" title="c1">L</font></span>)</span></span><br/>

<span class="kw">hence </span><a NAME="E4:21_1_1_2_2"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="lattice7.html#K3" title="LATTICE7:func.3">Join-IRR</a> <font color="Maroon" title="c1">L</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21_1_1_2_2"><span class="lab"><font color="Green" title="E26">A18</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E13:21_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="lattice7.html#K3" title="LATTICE7:func.3">Join-IRR</a> <font color="Maroon" title="c1">L</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"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="lattice7.html#K3" title="LATTICE7:func.3">Join-IRR</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="lattice7.html#K3" title="LATTICE7:func.3">Join-IRR</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:21_1_1_2"><span class="lab"><font color="Green" title="E22">A11</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/><a NAME="E14:21_1_1_2"/><span class="kw">then </span>
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="lattice7.html#K3" title="LATTICE7:func.3">Join-IRR</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><a href="lattice7.html#K3" title="LATTICE7:func.3">Join-IRR</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><a href="lattice7.html#K3" title="LATTICE7:func.3">Join-IRR</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:21_1_1_2"><span class="lab"><font color="Green" title="E23">A15</font></span></a>, <a class="txt" href="#E10:21_1_1_2"><span class="lab"><font color="Green" title="E24">A16</font></span></a>, <a class="ref" href="yellow_0.html#T36" title="YELLOW_0:th.36">YELLOW_0:36</a></span>;<br/><a NAME="E15:21_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A19</font></span>: 
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="lattice7.html#K3" title="LATTICE7:func.3">Join-IRR</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><a href="lattice7.html#K3" title="LATTICE7:func.3">Join-IRR</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="txt" href="#E7:21_1_1_2"><span class="lab"><font color="Green" title="E21">A10</font></span></a></span>;<br/><a NAME="E16:21_1_1_2"/>
<font color="Maroon" title="c4">z</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c2">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:21_1_1_2"><span class="lab"><font color="Green" title="E19">A8</font></span></a>, <a class="txt" href="#E11:21_1_1_2"><span class="lab"><font color="Green" title="E25">A17</font></span></a>, <a class="ref" href="orders_2.html#D10" title="ORDERS_2:def.10">ORDERS_2:def 10</a></span>;<br/><span class="kw">hence </span><a NAME="E17:21_1_1_2"/>
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="lattice7.html#K3" title="LATTICE7:func.3">Join-IRR</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="txt" href="#E3:21_1_1_2"><span class="lab"><font color="Green" title="E17">A6</font></span></a>, <a class="txt" href="#E15:21_1_1_2"><span class="lab"><font color="Green" title="E26">A19</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:21_1_1_3"/><span class="lab"><font color="Green" title="E17">A20</font></span>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lattice7.html#K3" title="LATTICE7:func.3">Join-IRR</a> <font color="Maroon" title="c1">L</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="lattice7.html#K3" title="LATTICE7:func.3">Join-IRR</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font></span><br/><div class="add"><a NAME="E2:21_1_1_3"/>
<font color="Maroon" title="c2">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">a</font>
 ;<br/><a NAME="E3:21_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c2">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_0.html#T17" title="WAYBEL_0:th.17">WAYBEL_0:17</a></span>;<br/><a NAME="E4:21_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="lattice7.html#K3" title="LATTICE7:func.3">Join-IRR</a> <font color="Maroon" title="c1">L</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_1_1_3"><span class="lab"><font color="Green" title="E17">A20</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E5:21_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A21</font></span>: 
<font color="Maroon" title="c2">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="lattice7.html#K3" title="LATTICE7:func.3">Join-IRR</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T17" title="YELLOW_0:th.17">YELLOW_0:17</a>, <a class="ref" href="yellow_4.html#T1" title="YELLOW_4:th.1">YELLOW_4:1</a></span>;<br/><a NAME="E6:21_1_1_3"/>
(  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="lattice7.html#K3" title="LATTICE7:func.3">Join-IRR</a> <font color="Maroon" title="c1">L</font></span>)</span>,<font color="Maroon" title="c1">L</font> &amp;  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a>  <a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c1">L</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T17" title="YELLOW_0:th.17">YELLOW_0:17</a></span>;<br/><a NAME="E7:21_1_1_3"/><span class="kw">then </span>
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="lattice7.html#K3" title="LATTICE7:func.3">Join-IRR</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c2">a</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a>, <a class="ref" href="yellow_0.html#T34" title="YELLOW_0:th.34">YELLOW_0:34</a></span>;<br/><a NAME="E8:21_1_1_3"/><span class="kw">then </span>
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="lattice7.html#K3" title="LATTICE7:func.3">Join-IRR</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_0.html#T34" title="WAYBEL_0:th.34">WAYBEL_0:34</a></span>;<br/><span class="kw">hence </span><a NAME="E9:21_1_1_3"/>
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="lattice7.html#K3" title="LATTICE7:func.3">Join-IRR</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:21_1_1_3"><span class="lab"><font color="Green" title="E18">A21</font></span></a>, <a class="ref" href="orders_2.html#T25" title="ORDERS_2:th.25">ORDERS_2:25</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>


</div>
