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

<span class="kw">let </span><font color="Maroon" title="c1">L</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="robbins3.html#L4" title="ROBBINS3:struct.4">OrthoLattRelStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">L</font> is  <a href="robbins3.html#V8" title="ROBBINS3:attr.8">involutive</a>  &amp; <font color="Maroon" title="c1">L</font> is  <a href="robbins3.html#V9" title="ROBBINS3:attr.9">with_Top</a>  &amp; <font color="Maroon" title="c1">L</font> is  <a href="robbins1.html#V10" title="ROBBINS1:attr.10">de_Morgan</a>  &amp; <font color="Maroon" title="c1">L</font> is  <a href="lattices.html#V10" title="LATTICES:attr.10">Lattice-like</a>  &amp; <font color="Maroon" title="c1">L</font> is  <a href="robbins3.html#V10" title="ROBBINS3:attr.10">naturally_sup-generated</a>  implies ( <font color="Maroon" title="c1">L</font> is  <a href="oposet_1.html#V12" title="OPOSET_1:attr.12">Orthocomplemented</a>  &amp; <font color="Maroon" title="c1">L</font> is  <a href="oposet_1.html#V6" title="OPOSET_1:attr.6">PartialOrdered</a>  ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:95"/>
( <font color="Maroon" title="c1">L</font> is  <a href="robbins3.html#V8" title="ROBBINS3:attr.8">involutive</a>  &amp; <font color="Maroon" title="c1">L</font> is  <a href="robbins3.html#V9" title="ROBBINS3:attr.9">with_Top</a>  &amp; <font color="Maroon" title="c1">L</font> is  <a href="robbins1.html#V10" title="ROBBINS1:attr.10">de_Morgan</a>  &amp; <font color="Maroon" title="c1">L</font> is  <a href="lattices.html#V10" title="LATTICES:attr.10">Lattice-like</a>  &amp; <font color="Maroon" title="c1">L</font> is  <a href="robbins3.html#V10" title="ROBBINS3:attr.10">naturally_sup-generated</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">L</font> is  <a href="oposet_1.html#V12" title="OPOSET_1:attr.12">Orthocomplemented</a>  &amp; <font color="Maroon" title="c1">L</font> is  <a href="oposet_1.html#V6" title="OPOSET_1:attr.6">PartialOrdered</a>  )</span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c2">L9</font> = <font color="Maroon" title="c1">L</font> as   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="lattices.html#V10" title="LATTICES:attr.10">Lattice-like</a>   <a href="robbins1.html#V10" title="ROBBINS1:attr.10">de_Morgan</a>   <a href="robbins3.html#V8" title="ROBBINS3:attr.8">involutive</a>   <a href="robbins3.html#V9" title="ROBBINS3:attr.9">with_Top</a>   <a href="robbins3.html#V10" title="ROBBINS3:attr.10">naturally_sup-generated</a>   <a href="robbins3.html#L4" title="ROBBINS3:struct.4">OrthoLattRelStr</a>  ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c3">f</font> =  the <a href="robbins1.html#U1" title="ROBBINS1:sel.1">Compl</a> of <font color="Maroon" title="c2">L9</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">L9</font>,<font color="Maroon" title="c2">L9</font> ;<br/>
<a NAME="E4:95"/>
 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 <font color="Maroon" title="c2">L9</font>  st <font color="Olive" title="b1">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b2">y</font> holds <br/><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L9</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">y</font> implies <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">y</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:95_1"/>
<font color="Maroon" title="c4">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">y</font></span><br/>

<a NAME="E2:95_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">x</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c5">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="robbins3.html#T22" target="_self" title="ROBBINS3:th.22">Th22</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E34">A1</font></span>: <font color="Maroon" title="c4">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a>  = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="robbins3.html#K4" title="ROBBINS3:func.4">|^|</a> <font color="Maroon" title="c5">y</font></span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#T4" title="LATTICES:th.4">LATTICES:4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a></span>)</span> <a href="robbins3.html#K5" title="ROBBINS3:func.5">|_|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a></span>)</span></span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a></span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="robbins1.html#D23" title="ROBBINS1:def.23">ROBBINS1:def 23</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a></span>)</span> <a href="robbins3.html#K5" title="ROBBINS3:func.5">|_|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="robbins3.html#D6" target="_self" title="ROBBINS3:def.6">Def6</a></span>
;<br/>
<a NAME="E4:95_1"/>
( <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a>  &amp; <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="robbins1.html#D3" title="ROBBINS1:def.3">ROBBINS1:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:95_1"/>
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:95_1"><span class="lab"><font color="Green" title="E34">A1</font></span></a>, <a class="ref" href="robbins3.html#D10" target="_self" title="ROBBINS3:def.10">Def10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E5:95"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A2</font></span>: 
<font color="Maroon" title="c3">f</font> is  <a href="waybel_0.html#V5" title="WAYBEL_0:attr.5">antitone</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_9.html#D1" title="WAYBEL_9:def.1">WAYBEL_9:def 1</a></span>;<br/>
<a NAME="E6:95"/><span class="lab"><font color="Green" title="E35">A3</font></span>: 
 for <font color="Olive" title="b1">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L9</font> holds <br/> (  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Olive" title="b1">y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">y</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Maroon" title="c2">L9</font> &amp;  <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Olive" title="b1">y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">y</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Maroon" title="c2">L9</font> &amp;  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> (<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Olive" title="b1">y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">y</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Maroon" title="c2">L9</font>) <a href="waybel_1.html#R4" title="WAYBEL_1:pred.4">is_maximum_of</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L9</font> &amp;  <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> (<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Olive" title="b1">y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">y</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Maroon" title="c2">L9</font>) <a href="waybel_1.html#R3" title="WAYBEL_1:pred.3">is_minimum_of</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L9</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c4">xx</font> =  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L9</font>,<font color="Maroon" title="c2">L9</font>);<br/>
<span class="kw">let </span><font color="Maroon" title="c5">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L9</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c5">y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">y</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Maroon" title="c2">L9</font> &amp;  <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c5">y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">y</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Maroon" title="c2">L9</font> &amp;  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> (<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c5">y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">y</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Maroon" title="c2">L9</font>) <a href="waybel_1.html#R4" title="WAYBEL_1:pred.4">is_maximum_of</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L9</font> &amp;  <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> (<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c5">y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">y</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Maroon" title="c2">L9</font>) <a href="waybel_1.html#R3" title="WAYBEL_1:pred.3">is_minimum_of</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L9</font> )</span><br/>

<span class="kw">thus </span><a NAME="E1:95_2"/>
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c5">y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">y</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Maroon" title="c2">L9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T20" title="YELLOW_0:th.20">YELLOW_0:20</a></span>; <a class="txt" 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="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c5">y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">y</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Maroon" title="c2">L9</font> &amp;  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> (<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c5">y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">y</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Maroon" title="c2">L9</font>) <a href="waybel_1.html#R4" title="WAYBEL_1:pred.4">is_maximum_of</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L9</font> &amp;  <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> (<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c5">y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">y</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Maroon" title="c2">L9</font>) <a href="waybel_1.html#R3" title="WAYBEL_1:pred.3">is_minimum_of</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L9</font> )</span><br/>

<span class="kw">thus </span><a NAME="E2:95_2"/>
 <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c5">y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">y</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Maroon" title="c2">L9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T21" title="YELLOW_0:th.21">YELLOW_0:21</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> (<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c5">y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">y</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Maroon" title="c2">L9</font>) <a href="waybel_1.html#R4" title="WAYBEL_1:pred.4">is_maximum_of</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L9</font> &amp;  <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> (<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c5">y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">y</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Maroon" title="c2">L9</font>) <a href="waybel_1.html#R3" title="WAYBEL_1:pred.3">is_minimum_of</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L9</font> )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c6">t</font> = <font color="Maroon" title="c5">y</font> <a href="robbins3.html#K5" title="ROBBINS3:func.5">|_|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a></span>)</span>;<br/>
<a NAME="E3:95_2"/>
 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="c2">L9</font>  st <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L9</font> holds <br/><font color="Olive" title="b1">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">y</font> <a href="robbins3.html#K5" title="ROBBINS3:func.5">|_|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L9</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L9</font> implies <font color="Maroon" title="c7">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">y</font> <a href="robbins3.html#K5" title="ROBBINS3:func.5">|_|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:95_2_1"/>
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">y</font> <a href="robbins3.html#K5" title="ROBBINS3:func.5">|_|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a></span>)</span></span><br/>

<font color="Maroon" title="c7">b</font> <a href="robbins3.html#K5" title="ROBBINS3:func.5">|_|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="robbins3.html#K5" title="ROBBINS3:func.5">|_|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a></span>)</span></span>)</span> = 
<font color="Maroon" title="c7">b</font> <a href="robbins3.html#K5" title="ROBBINS3:func.5">|_|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">b</font> <a href="robbins3.html#K5" title="ROBBINS3:func.5">|_|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">b</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="robbins3.html#D7" target="_self" title="ROBBINS3:def.7">Def7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">b</font> <a href="robbins3.html#K5" title="ROBBINS3:func.5">|_|</a> <font color="Maroon" title="c7">b</font></span>)</span> <a href="robbins3.html#K5" title="ROBBINS3:func.5">|_|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">b</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#D5" title="LATTICES:def.5">LATTICES:def 5</a></span>
<br/>.= 
<font color="Maroon" title="c7">b</font> <a href="robbins3.html#K5" title="ROBBINS3:func.5">|_|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">b</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#T1" title="LATTICES:th.1">LATTICES:1</a></span>
<br/>.= 
<font color="Maroon" title="c5">y</font> <a href="robbins3.html#K5" title="ROBBINS3:func.5">|_|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="robbins3.html#D7" target="_self" title="ROBBINS3:def.7">Def7</a></span>
;<br/>
<a NAME="E3:95_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">b</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c5">y</font> <a href="robbins3.html#K5" title="ROBBINS3:func.5">|_|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#D3" title="LATTICES:def.3">LATTICES:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:95_2_1"/>
<font color="Maroon" title="c7">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">y</font> <a href="robbins3.html#K5" title="ROBBINS3:func.5">|_|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="robbins3.html#T22" target="_self" title="ROBBINS3:th.22">Th22</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E4:95_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A4</font></span>: 
<font color="Maroon" title="c5">y</font> <a href="robbins3.html#K5" title="ROBBINS3:func.5">|_|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a></span>)</span> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#D9" title="LATTICE3:def.9">LATTICE3:def 9</a></span>;<br/>
<a NAME="E5:95_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">L9</font> is  <a href="yellow_0.html#V2" title="YELLOW_0:attr.2">upper-bounded</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#D5" title="YELLOW_0:def.5">YELLOW_0:def 5</a></span>;<br/>
<a NAME="E6:95_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A5</font></span>: 
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L9</font>,<font color="Maroon" title="c2">L9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T43" title="YELLOW_0:th.43">YELLOW_0:43</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">t</font> = <font color="Maroon" title="c5">y</font> <a href="robbins3.html#K5" title="ROBBINS3:func.5">|_|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a></span>)</span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L9</font> ;<br/>
<a NAME="E8:95_2"/><span class="lab"><font color="Green" title="E38">A6</font></span>: 
 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="c2">L9</font>  st  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L9</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Olive" title="b1">a</font> holds <br/><font color="Maroon" title="c7">t</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#D9" title="LATTICE3:def.9">LATTICE3:def 9</a></span>;<br/>
 <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> (<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c5">y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">y</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Maroon" title="c2">L9</font>) = 
 <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> (<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c5">y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Maroon" title="c2">L9</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="robbins1.html#D3" title="ROBBINS1:def.3">ROBBINS1:def 3</a></span>
<br/>.= 
<font color="Maroon" title="c5">y</font> <a href="robbins3.html#NK4" title="ROBBINS3:NK.4">"|_|"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T41" title="YELLOW_0:th.41">YELLOW_0:41</a></span>
<br/>.= 
<font color="Maroon" title="c5">y</font> <a href="robbins3.html#K5" title="ROBBINS3:func.5">|_|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="robbins3.html#T25" target="_self" title="ROBBINS3:th.25">Th25</a></span>
<br/>.= 
 <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L9</font>,<font color="Maroon" title="c2">L9</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:95_2"><span class="lab"><font color="Green" title="E36">A4</font></span></a>, <a class="txt" href="#E6:95_2"><span class="lab"><font color="Green" title="E37">A5</font></span></a>, <a class="txt" href="#E8:95_2"><span class="lab"><font color="Green" title="E38">A6</font></span></a>, <a class="ref" href="yellow_0.html#D9" title="YELLOW_0:def.9">YELLOW_0:def 9</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E10:95_2"/>
 <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> (<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c5">y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">y</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Maroon" title="c2">L9</font>) <a href="waybel_1.html#R4" title="WAYBEL_1:pred.4">is_maximum_of</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:95_2"><span class="lab"><font color="Green" title="E37">A5</font></span></a>, <a class="ref" href="waybel_1.html#D7" title="WAYBEL_1:def.7">WAYBEL_1:def 7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> (<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c5">y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">y</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Maroon" title="c2">L9</font>) <a href="waybel_1.html#R3" title="WAYBEL_1:pred.3">is_minimum_of</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L9</font></span><br/>

<span class="kw">set </span><font color="Maroon" title="c8">xx</font> =  <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L9</font>,<font color="Maroon" title="c2">L9</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c9">t</font> = <font color="Maroon" title="c5">y</font> <a href="robbins3.html#K4" title="ROBBINS3:func.4">|^|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a></span>)</span>;<br/>
<a NAME="E11:95_2"/><span class="lab"><font color="Green" title="E39">A7</font></span>: 
 for <font color="Olive" title="b1">a</font>, <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="c2">L9</font> holds  <font color="Olive" title="b1">a</font> <a href="robbins3.html#K4" title="ROBBINS3:func.4">|^|</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> <a href="robbins3.html#K4" title="ROBBINS3:func.4">|^|</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">b</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_2_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">a</font>, <font color="Maroon" title="c11">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L9</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">a</font> <a href="robbins3.html#K4" title="ROBBINS3:func.4">|^|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">b</font> <a href="robbins3.html#K4" title="ROBBINS3:func.4">|^|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">b</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a></span>)</span></span><br/>

<font color="Maroon" title="c10">a</font> <a href="robbins3.html#K4" title="ROBBINS3:func.4">|^|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a></span>)</span> <a href="robbins3.html#K5" title="ROBBINS3:func.5">|_|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a></span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a></span>)</span></span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="robbins1.html#D23" title="ROBBINS1:def.23">ROBBINS1:def 23</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">b</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a></span>)</span> <a href="robbins3.html#K5" title="ROBBINS3:func.5">|_|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">b</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a></span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a></span>)</span></span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="robbins3.html#D7" target="_self" title="ROBBINS3:def.7">Def7</a></span>
<br/>.= 
<font color="Maroon" title="c11">b</font> <a href="robbins3.html#K4" title="ROBBINS3:func.4">|^|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">b</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="robbins1.html#D23" title="ROBBINS1:def.23">ROBBINS1:def 23</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E2:95_2_3"/>
<font color="Maroon" title="c10">a</font> <a href="robbins3.html#K4" title="ROBBINS3:func.4">|^|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">b</font> <a href="robbins3.html#K4" title="ROBBINS3:func.4">|^|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">b</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E12:95_2"/>
 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="c2">L9</font>  st <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L9</font> holds <br/><font color="Olive" title="b1">b</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c5">y</font> <a href="robbins3.html#K4" title="ROBBINS3:func.4">|^|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_2_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L9</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L9</font> implies <font color="Maroon" title="c10">b</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c5">y</font> <a href="robbins3.html#K4" title="ROBBINS3:func.4">|^|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:95_2_4"/>
<font color="Maroon" title="c10">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">b</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c5">y</font> <a href="robbins3.html#K4" title="ROBBINS3:func.4">|^|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a></span>)</span></span><br/>

<font color="Maroon" title="c10">b</font> <a href="robbins3.html#K4" title="ROBBINS3:func.4">|^|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="robbins3.html#K4" title="ROBBINS3:func.4">|^|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a></span>)</span></span>)</span> = 
<font color="Maroon" title="c10">b</font> <a href="robbins3.html#K4" title="ROBBINS3:func.4">|^|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">b</font> <a href="robbins3.html#K4" title="ROBBINS3:func.4">|^|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">b</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:95_2"><span class="lab"><font color="Green" title="E39">A7</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">b</font> <a href="robbins3.html#K4" title="ROBBINS3:func.4">|^|</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="robbins3.html#K4" title="ROBBINS3:func.4">|^|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">b</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#D7" title="LATTICES:def.7">LATTICES:def 7</a></span>
<br/>.= 
<font color="Maroon" title="c10">b</font> <a href="robbins3.html#K4" title="ROBBINS3:func.4">|^|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">b</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#T2" title="LATTICES:th.2">LATTICES:2</a></span>
<br/>.= 
<font color="Maroon" title="c5">y</font> <a href="robbins3.html#K4" title="ROBBINS3:func.4">|^|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:95_2"><span class="lab"><font color="Green" title="E39">A7</font></span></a></span>
;<br/>
<a NAME="E3:95_2_4"/><span class="kw">then </span>
<font color="Maroon" title="c5">y</font> <a href="robbins3.html#K4" title="ROBBINS3:func.4">|^|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a></span>)</span> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c10">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#T4" title="LATTICES:th.4">LATTICES:4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:95_2_4"/>
<font color="Maroon" title="c10">b</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c5">y</font> <a href="robbins3.html#K4" title="ROBBINS3:func.4">|^|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="robbins3.html#T22" target="_self" title="ROBBINS3:th.22">Th22</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E13:95_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A8</font></span>: 
<font color="Maroon" title="c5">y</font> <a href="robbins3.html#K4" title="ROBBINS3:func.4">|^|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a></span>)</span> <a href="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#D8" title="LATTICE3:def.8">LATTICE3:def 8</a></span>;<br/>
<a NAME="E14:95_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">L9</font> is  <a href="yellow_0.html#V1" title="YELLOW_0:attr.1">lower-bounded</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#D4" title="YELLOW_0:def.4">YELLOW_0:def 4</a></span>;<br/>
<a NAME="E15:95_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A9</font></span>: 
 <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L9</font>,<font color="Maroon" title="c2">L9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T42" title="YELLOW_0:th.42">YELLOW_0:42</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">t</font> = <font color="Maroon" title="c5">y</font> <a href="robbins3.html#K4" title="ROBBINS3:func.4">|^|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a></span>)</span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L9</font> ;<br/>
<a NAME="E17:95_2"/><span class="lab"><font color="Green" title="E42">A10</font></span>: 
 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="c2">L9</font>  st  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L9</font> <a href="lattice3.html#NR3" title="LATTICE3:NR.3">is_&gt;=_than</a> <font color="Olive" title="b1">a</font> holds <br/><font color="Maroon" title="c10">t</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Olive" title="b1">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#D8" title="LATTICE3:def.8">LATTICE3:def 8</a></span>;<br/>
 <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> (<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c5">y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">y</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Maroon" title="c2">L9</font>) = 
 <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> (<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c5">y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Maroon" title="c2">L9</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="robbins1.html#D3" title="ROBBINS1:def.3">ROBBINS1:def 3</a></span>
<br/>.= 
<font color="Maroon" title="c5">y</font> <a href="robbins3.html#NK3" title="ROBBINS3:NK.3">"|^|"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T40" title="YELLOW_0:th.40">YELLOW_0:40</a></span>
<br/>.= 
<font color="Maroon" title="c5">y</font> <a href="robbins3.html#K4" title="ROBBINS3:func.4">|^|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="robbins3.html#T26" target="_self" title="ROBBINS3:th.26">Th26</a></span>
<br/>.= 
 <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L9</font>,<font color="Maroon" title="c2">L9</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:95_2"><span class="lab"><font color="Green" title="E40">A8</font></span></a>, <a class="txt" href="#E15:95_2"><span class="lab"><font color="Green" title="E41">A9</font></span></a>, <a class="txt" href="#E17:95_2"><span class="lab"><font color="Green" title="E42">A10</font></span></a>, <a class="ref" href="yellow_0.html#D10" title="YELLOW_0:def.10">YELLOW_0:def 10</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E19:95_2"/>
 <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> (<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c5">y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">y</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Maroon" title="c2">L9</font>) <a href="waybel_1.html#R3" title="WAYBEL_1:pred.3">is_minimum_of</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:95_2"><span class="lab"><font color="Green" title="E41">A9</font></span></a>, <a class="ref" href="waybel_1.html#D6" title="WAYBEL_1:def.6">WAYBEL_1:def 6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E7:95"/>
 for <font color="Olive" title="b1">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L9</font> holds  <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L9</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font></span><br/>

<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font></span>)</span> = 
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="robbins1.html#D3" title="ROBBINS1:def.3">ROBBINS1:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a></span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="robbins1.html#D3" title="ROBBINS1:def.3">ROBBINS1:def 3</a></span>
<br/>.= 
<font color="Maroon" title="c4">x</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="robbins3.html#D6" target="_self" title="ROBBINS3:def.6">Def6</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E2:95_3"/>
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E8:95"/><span class="kw">then </span>
<font color="Maroon" title="c3">f</font> is  <a href="partit_2.html#V1" title="PARTIT_2:attr.1">involutive</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partit_2.html#D3" title="PARTIT_2:def.3">PARTIT_2:def 3</a></span>;<br/>
<a NAME="E9:95"/><span class="kw">then </span>
<font color="Maroon" title="c3">f</font> is  <a href="oposet_1.html#V9" title="OPOSET_1:attr.9">Orderinvolutive</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:95"><span class="lab"><font color="Green" title="E34">A2</font></span></a>, <a class="ref" href="oposet_1.html#D17" title="OPOSET_1:def.17">OPOSET_1:def 17</a></span>;<br/>
<a NAME="E10:95"/><span class="kw">then </span>
<font color="Maroon" title="c3">f</font> <a href="oposet_1.html#R2" title="OPOSET_1:pred.2">OrthoComplement_on</a> <font color="Maroon" title="c2">L9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:95"><span class="lab"><font color="Green" title="E35">A3</font></span></a>, <a class="ref" href="oposet_1.html#D21" title="OPOSET_1:def.21">OPOSET_1:def 21</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:95"/>
( <font color="Maroon" title="c1">L</font> is  <a href="oposet_1.html#V12" title="OPOSET_1:attr.12">Orthocomplemented</a>  &amp; <font color="Maroon" title="c1">L</font> is  <a href="oposet_1.html#V6" title="OPOSET_1:attr.6">PartialOrdered</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="oposet_1.html#D22" title="OPOSET_1:def.22">OPOSET_1:def 22</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
