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

<span class="kw">let </span><font color="Maroon" title="c1">R</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="lfuzzy_0.html#V2" title="LFUZZY_0:attr.2">interval</a>   <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</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">R</font> is  <a href="waybel_1.html#V9" title="WAYBEL_1:attr.9">Heyting</a> </span><br/>

<span class="kw">thus </span><a NAME="E1:25_1_1"/>
<font color="Maroon" title="c1">R</font> is   <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="waybel_1.html#D19" title="WAYBEL_1:def.19">WAYBEL_1:def 19</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive">b<sub>1</sub></font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> holds  <font color="Olive">b<sub>1</sub></font> <a href="waybel_1.html#K4" title="WAYBEL_1:func.4">"/\"</a>  is  <a href="waybel_1.html#V5" title="WAYBEL_1:attr.5">lower_adjoint</a> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c2">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</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="c2">x</font> <a href="waybel_1.html#K4" title="WAYBEL_1:func.4">"/\"</a>  is  <a href="waybel_1.html#V5" title="WAYBEL_1:attr.5">lower_adjoint</a> </span><br/>

<span class="kw">set </span><font color="Maroon" title="c3">f</font> = <font color="Maroon" title="c2">x</font> <a href="waybel_1.html#K4" title="WAYBEL_1:func.4">"/\"</a> ;<br/>
<a NAME="E2:25_1_1"/>
<font color="Maroon" title="c2">x</font> <a href="waybel_1.html#K4" title="WAYBEL_1:func.4">"/\"</a>  is  <a href="waybel_0.html#V18" title="WAYBEL_0:attr.18">sups-preserving</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_1"><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#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">R</font>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="waybel_0.html#D33" title="WAYBEL_0:def.33">WAYBEL_0:def 33</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">x</font> <a href="waybel_1.html#K4" title="WAYBEL_1:func.4">"/\"</a>  <a href="waybel_0.html#R4" title="WAYBEL_0:pred.4">preserves_sup_of</a> <font color="Maroon" title="c4">X</font></span><br/>

<span class="kw">assume </span><a NAME="E1:25_1_1_1"/>
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">R</font>
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="waybel_0.html#D31" title="WAYBEL_0:def.31">WAYBEL_0:def 31</a> <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">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="waybel_1.html#K4" title="WAYBEL_1:func.4">"/\"</a></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">R</font> &amp;  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="waybel_1.html#K4" title="WAYBEL_1:func.4">"/\"</a></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c4">X</font></span>)</span>,<font color="Maroon" title="c1">R</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="waybel_1.html#K4" title="WAYBEL_1:func.4">"/\"</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">R</font>)</span>)</span> )</span><br/>

<div><span class="lab"><font color="Green" title="E10">A1</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</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="c5">b</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="waybel_1.html#K4" title="WAYBEL_1:func.4">"/\"</a></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c4">X</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="waybel_1.html#K4" title="WAYBEL_1:func.4">"/\"</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="lfuzzy_0.html#NR1" title="LFUZZY_0:NR.1">&lt;&lt;=</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><span class="kw">assume </span><a NAME="E1:25_1_1_1_1"/><span class="lab"><font color="Green" title="E11">A2</font></span>: 
<font color="Maroon" title="c5">b</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="waybel_1.html#K4" title="WAYBEL_1:func.4">"/\"</a></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</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"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="waybel_1.html#K4" title="WAYBEL_1:func.4">"/\"</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="lfuzzy_0.html#NR1" title="LFUZZY_0:NR.1">&lt;&lt;=</a> <font color="Olive">b<sub>1</sub></font></span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_1_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="E1:25_1_1_1_1_1"/>
( <font color="Maroon" title="c2">x</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a> <font color="Maroon" title="c4">X</font> or  not <font color="Maroon" title="c2">x</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a> <font color="Maroon" title="c4">X</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:25_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E12">A3</font></span>: 
<font color="Maroon" title="c2">x</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</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"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="waybel_1.html#K4" title="WAYBEL_1:func.4">"/\"</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="lfuzzy_0.html#NR1" title="LFUZZY_0:NR.1">&lt;&lt;=</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><a NAME="E2:25_1_1_1_1_1_1"/>
<font color="Maroon" title="c5">b</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a> <font color="Maroon" title="c4">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_1_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="lattice3.html#D9" title="LATTICE3:def.9">LATTICE3:def 9</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> or <font color="Maroon" title="c6">a</font> <a href="lfuzzy_0.html#NR1" title="LFUZZY_0:NR.1">&lt;&lt;=</a> <font color="Maroon" title="c5">b</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:25_1_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E13">A4</font></span>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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"> <font color="Maroon" title="c6">a</font> <a href="lfuzzy_0.html#NR1" title="LFUZZY_0:NR.1">&lt;&lt;=</a> <font color="Maroon" title="c5">b</font></span><br/>

<a NAME="E2:25_1_1_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">x</font> <a href="lfuzzy_0.html#NR2" title="LFUZZY_0:NR.2">&gt;&gt;=</a> <font color="Maroon" title="c6">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:25_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E12">A3</font></span></a>, <a class="ref" href="lattice3.html#D9" title="LATTICE3:def.9">LATTICE3:def 9</a></span>;<br/>
<a NAME="E3:25_1_1_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</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="E4:25_1_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E14">A5</font></span>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="waybel_1.html#K4" title="WAYBEL_1:func.4">"/\"</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_1.html#D18" title="WAYBEL_1:def.18">WAYBEL_1:def 18</a></span>;<br/>
<a NAME="E5:25_1_1_1_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="waybel_1.html#K4" title="WAYBEL_1:func.4">"/\"</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="waybel_1.html#K4" title="WAYBEL_1:func.4">"/\"</a></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c4">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:25_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E13">A4</font></span></a>, <a class="ref" href="funct_2.html#T35" title="FUNCT_2:th.35">FUNCT_2:35</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:25_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c6">a</font> <a href="lfuzzy_0.html#NR1" title="LFUZZY_0:NR.1">&lt;&lt;=</a> <font color="Maroon" title="c5">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:25_1_1_1_1"><span class="lab"><font color="Green" title="E11">A2</font></span></a>, <a class="txt" href="#E4:25_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E14">A5</font></span></a>, <a class="ref" href="lattice3.html#D9" title="LATTICE3:def.9">LATTICE3:def 9</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="E3:25_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E13">A6</font></span>: 
<font color="Maroon" title="c5">b</font> <a href="lfuzzy_0.html#NR2" title="LFUZZY_0:NR.2">&gt;&gt;=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c4">X</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T32" title="YELLOW_0:th.32">YELLOW_0:32</a></span>;<br/><a NAME="E4:25_1_1_1_1_1_1"/>
<font color="Maroon" title="c2">x</font> <a href="lfuzzy_0.html#NR2" title="LFUZZY_0:NR.2">&gt;&gt;=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c4">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:25_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E12">A3</font></span></a>, <a class="ref" href="yellow_0.html#T32" title="YELLOW_0:th.32">YELLOW_0:32</a></span>;<br/><a NAME="E5:25_1_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c4">X</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/><span class="kw">hence </span><a NAME="E6:25_1_1_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="waybel_1.html#K4" title="WAYBEL_1:func.4">"/\"</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="lfuzzy_0.html#NR1" title="LFUZZY_0:NR.1">&lt;&lt;=</a> <font color="Maroon" title="c5">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:25_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E13">A6</font></span></a>, <a class="ref" href="waybel_1.html#D18" title="WAYBEL_1:def.18">WAYBEL_1:def 18</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:25_1_1_1_1_1_2"/>
 not <font color="Maroon" title="c2">x</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</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"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="waybel_1.html#K4" title="WAYBEL_1:func.4">"/\"</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="lfuzzy_0.html#NR1" title="LFUZZY_0:NR.1">&lt;&lt;=</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font><span class="kw"> such that </span><br/><a NAME="E3:25_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E12">A7</font></span>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 <span class="kw">and </span><br/><a NAME="E4:25_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E13">A8</font></span>: 
 not <font color="Maroon" title="c2">x</font> <a href="lfuzzy_0.html#NR2" title="LFUZZY_0:NR.2">&gt;&gt;=</a> <font color="Maroon" title="c6">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 NAME="E5:25_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E14">A9</font></span>: 
<font color="Maroon" title="c2">x</font> <a href="lfuzzy_0.html#NR1" title="LFUZZY_0:NR.1">&lt;&lt;=</a> <font color="Maroon" title="c6">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:25_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E13">A8</font></span></a>, <a class="ref" href="waybel_0.html#D29" title="WAYBEL_0:def.29">WAYBEL_0:def 29</a></span>;<br/><a NAME="E6:25_1_1_1_1_1_2"/>
<font color="Maroon" title="c6">a</font> <a href="lfuzzy_0.html#NR1" title="LFUZZY_0:NR.1">&lt;&lt;=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c4">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:25_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E12">A7</font></span></a>, <a class="ref" href="yellow_2.html#T22" title="YELLOW_2:th.22">YELLOW_2:22</a></span>;<br/><a NAME="E7:25_1_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">x</font> <a href="lfuzzy_0.html#NR1" title="LFUZZY_0:NR.1">&lt;&lt;=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c4">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:25_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E14">A9</font></span></a>, <a class="ref" href="orders_2.html#T3" title="ORDERS_2:th.3">ORDERS_2:3</a></span>;<br/><a NAME="E8:25_1_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c4">X</font></span>)</span>
 <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="E9:25_1_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A10</font></span>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="waybel_1.html#K4" title="WAYBEL_1:func.4">"/\"</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c4">X</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_1.html#D18" title="WAYBEL_1:def.18">WAYBEL_1:def 18</a></span>;<br/><a NAME="E10:25_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E16">A11</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="waybel_1.html#K4" title="WAYBEL_1:func.4">"/\"</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="waybel_1.html#K4" title="WAYBEL_1:func.4">"/\"</a></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c4">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:25_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E12">A7</font></span></a>, <a class="ref" href="funct_2.html#T35" title="FUNCT_2:th.35">FUNCT_2:35</a></span>;<br/><font color="Maroon" title="c2">x</font> = 
<font color="Maroon" title="c2">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c6">a</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:25_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E14">A9</font></span></a>, <a class="ref" href="yellow_0.html#T25" title="YELLOW_0:th.25">YELLOW_0:25</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="waybel_1.html#K4" title="WAYBEL_1:func.4">"/\"</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">a</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_1.html#D18" title="WAYBEL_1:def.18">WAYBEL_1:def 18</a></span>
;<br/><span class="kw">hence </span><a NAME="E12:25_1_1_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="waybel_1.html#K4" title="WAYBEL_1:func.4">"/\"</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="lfuzzy_0.html#NR1" title="LFUZZY_0:NR.1">&lt;&lt;=</a> <font color="Maroon" title="c5">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:25_1_1_1_1"><span class="lab"><font color="Green" title="E11">A2</font></span></a>, <a class="txt" href="#E9:25_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E15">A10</font></span></a>, <a class="txt" href="#E10:25_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E16">A11</font></span></a>, <a class="ref" href="lattice3.html#D9" title="LATTICE3:def.9">LATTICE3:def 9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">thus </span><a NAME="E3:25_1_1_1"/>
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="waybel_1.html#K4" title="WAYBEL_1:func.4">"/\"</a></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">R</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lfuzzy_0.html#T8" target="_self" title="LFUZZY_0:th.8">Th8</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">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="waybel_1.html#K4" title="WAYBEL_1:func.4">"/\"</a></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c4">X</font></span>)</span>,<font color="Maroon" title="c1">R</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="waybel_1.html#K4" title="WAYBEL_1:func.4">"/\"</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">R</font>)</span>)</span></span><br/>

<a NAME="E4:25_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="waybel_1.html#K4" title="WAYBEL_1:func.4">"/\"</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="waybel_1.html#K4" title="WAYBEL_1:func.4">"/\"</a></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c4">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="lattice3.html#D9" title="LATTICE3:def.9">LATTICE3:def 9</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="waybel_1.html#K4" title="WAYBEL_1:func.4">"/\"</a></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c4">X</font> or <font color="Maroon" title="c5">a</font> <a href="lfuzzy_0.html#NR1" title="LFUZZY_0:NR.1">&lt;&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="waybel_1.html#K4" title="WAYBEL_1:func.4">"/\"</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c4">X</font></span>)</span> )</span><br/>

<a NAME="E1:25_1_1_1_2"/><span class="lab"><font color="Green" title="E11">A12</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="waybel_1.html#K4" title="WAYBEL_1:func.4">"/\"</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c4">X</font></span>)</span> &amp; <font color="Maroon" title="c2">x</font> <a href="lfuzzy_0.html#NR1" title="LFUZZY_0:NR.1">&lt;&lt;=</a> <font color="Maroon" title="c2">x</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_1.html#D18" title="WAYBEL_1:def.18">WAYBEL_1:def 18</a>, <a class="ref" href="yellow_0.html#D1" title="YELLOW_0:def.1">YELLOW_0:def 1</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:25_1_1_1_2"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="waybel_1.html#K4" title="WAYBEL_1:func.4">"/\"</a></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</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"> <font color="Maroon" title="c5">a</font> <a href="lfuzzy_0.html#NR1" title="LFUZZY_0:NR.1">&lt;&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="waybel_1.html#K4" title="WAYBEL_1:func.4">"/\"</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c4">X</font></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:25_1_1_1_2"/><span class="lab"><font color="Green" title="E12">A13</font></span>: 
<font color="Maroon" title="c6">y</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="c1">R</font>
 <span class="kw">and </span><br/><a NAME="E5:25_1_1_1_2"/><span class="lab"><font color="Green" title="E13">A14</font></span>: 
( <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> &amp; <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="waybel_1.html#K4" title="WAYBEL_1:func.4">"/\"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">y</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T64" title="FUNCT_2:th.64">FUNCT_2:64</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">y</font> = <font color="Maroon" title="c6">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:25_1_1_1_2"><span class="lab"><font color="Green" title="E12">A13</font></span></a></span>;<br/>
<a NAME="E7:25_1_1_1_2"/>
( <font color="Maroon" title="c7">y</font> <a href="lfuzzy_0.html#NR1" title="LFUZZY_0:NR.1">&lt;&lt;=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c4">X</font> &amp; <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c7">y</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:25_1_1_1_2"><span class="lab"><font color="Green" title="E13">A14</font></span></a>, <a class="ref" href="waybel_1.html#D18" title="WAYBEL_1:def.18">WAYBEL_1:def 18</a>, <a class="ref" href="yellow_2.html#T22" title="YELLOW_2:th.22">YELLOW_2:22</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:25_1_1_1_2"/>
<font color="Maroon" title="c5">a</font> <a href="lfuzzy_0.html#NR1" title="LFUZZY_0:NR.1">&lt;&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="waybel_1.html#K4" title="WAYBEL_1:func.4">"/\"</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c4">X</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:25_1_1_1_2"><span class="lab"><font color="Green" title="E11">A12</font></span></a>, <a class="ref" href="yellow_3.html#T2" title="YELLOW_3:th.2">YELLOW_3:2</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>
<span class="kw">hence </span><a NAME="E5:25_1_1_1"/>
 <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="waybel_1.html#K4" title="WAYBEL_1:func.4">"/\"</a></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c4">X</font></span>)</span>,<font color="Maroon" title="c1">R</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="waybel_1.html#K4" title="WAYBEL_1:func.4">"/\"</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">R</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:25_1_1_1"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="ref" href="yellow_0.html#T30" title="YELLOW_0:th.30">YELLOW_0:30</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>
<span class="kw">hence </span><a NAME="E3:25_1_1"/>
<font color="Maroon" title="c2">x</font> <a href="waybel_1.html#K4" title="WAYBEL_1:func.4">"/\"</a>  is  <a href="waybel_1.html#V5" title="WAYBEL_1:attr.5">lower_adjoint</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_1.html#T17" title="WAYBEL_1:th.17">WAYBEL_1:17</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>
