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

<b>let </b><font color="Maroon">S</font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel_0.html#NM8">LATTICE</a>, <font color="Maroon">T</font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/><b>let </b><font color="Maroon">g</font> be  <a href="waybel_0.html#V17">infs-preserving</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">S</font>,<font color="Maroon">T</font>;<br/>



<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:38_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">g</font> is <a href="waybel_0.html#V22">directed-sups-preserving</a>
 ;<br/><b>let </b><font color="Maroon">X</font> be  <a href="waybel11.html#V4">Scott</a>  <a href="yellow_9.html#M1">TopAugmentation</a> of <font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">Y</font> be  <a href="waybel11.html#V4">Scott</a>  <a href="yellow_9.html#M1">TopAugmentation</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">V</font> be  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/><a NAME="E2:38_1"/><i><font color="Green">E45</font></i>: 
(  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">X</font> #) <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">T</font> #) &amp;  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">Y</font> #) <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</font> #) )
 <b>by </b><i><a class="ref" href="yellow_9.html#D4">YELLOW_9:def 4</a></i>;<br/><b>then reconsider </b><font color="Maroon">g'</font> = <font color="Maroon">g</font> as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">Y</font>,<font color="Maroon">X</font> ;<br/><b>reconsider </b><font color="Maroon">d</font> =  <a href="waybel34.html#K1" target="_self">LowerAdj</a> <font color="Maroon">g</font> as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font> <b>by </b><i/>;<br/><a NAME="E5:38_1"/>
 <a href="waybel_0.html#K4">uparrow</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">V</font></span>)</span> is <a href="waybel11.html#V1">inaccessible_by_directed_joins</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">D</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V1">directed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel11.html#D1">WAYBEL11:def 1</a><br/>

<b>assume </b><a NAME="E1:38_1_1"/>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K4">uparrow</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">V</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">y</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E3:38_1_1"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">d</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">V</font> )
 <b>by </b><i><a class="ref" href="waybel_0.html#D16">WAYBEL_0:def 16</a></i>;<br/>
<b>consider </b><font color="Maroon">u</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:38_1_1"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font> )
 <b>by </b><i>, <a class="ref" href="funct_2.html#T115">FUNCT_2:115</a></i>;<br/>
<b>reconsider </b><font color="Maroon">u</font> = <font color="Maroon">u</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">g</font> = <font color="Maroon">g'</font> as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">Y</font>,<font color="Maroon">X</font> ;<br/>
<a NAME="E8:38_1_1"/>
<span class="p1"><a href="waybel_1.html#K1">[</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">d</font></span><a href="waybel_1.html#K1">]</a></span> is  <a href="waybel_1.html#V3">Galois</a>  <a href="waybel_1.html#M1">Connection</a> of <font color="Maroon">S</font>,<font color="Maroon">T</font>
 
<b>by </b><i/>;<br/>
<a NAME="E9:38_1_1"/><b>then </b>
<span class="p1"><a href="waybel_1.html#K1">[</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">d</font></span><a href="waybel_1.html#K1">]</a></span> is <a href="waybel_1.html#V3">Galois</a>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E10:38_1_1"/><b>then </b><i><font color="Green">E48</font></i>: 
( <font color="Maroon">d</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">g</font> <a href="yellow_2.html#R1">&lt;=</a>  <a href="grcat_1.html#K7">id</a> <font color="Maroon">Y</font> &amp;  <a href="grcat_1.html#K7">id</a> <font color="Maroon">X</font> <a href="yellow_2.html#R1">&lt;=</a> <font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">d</font> )
 
<b>by </b><i><a class="ref" href="waybel_1.html#T19">WAYBEL_1:19</a></i>;<br/>
<a NAME="E11:38_1_1"/><i><font color="Green">E88</font></i>: 
( <span class="p1">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">X</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">d</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a>, <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>;<br/>
<a NAME="E12:38_1_1"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">g</font> is  <a href="waybel_0.html#V17">infs-preserving</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">Y</font>,<font color="Maroon">X</font>
 
<b>by </b><i>, <a class="ref" href="waybel21.html#T6">WAYBEL21:6</a></i>;<br/>
<a NAME="E13:38_1_1"/><b>then </b>
( <font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> &amp; <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D</font></span>)</span> )
 
<b>by </b><i>, , , , <a class="ref" href="orders_3.html#D5">ORDERS_3:def 5</a>, <a class="ref" href="yellow_2.html#T10">YELLOW_2:10</a></i>;<br/>
<a NAME="E14:38_1_1"/><b>then </b>
( <font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D</font></span>)</span> &amp; <font color="Maroon">V</font> is <a href="waybel_0.html#V13">upper</a> )
 
<b>by </b><i><a class="ref" href="orders_2.html#T26">ORDERS_2:26</a>, <a class="ref" href="waybel11.html#D4">WAYBEL11:def 4</a></i>;<br/>
<a NAME="E15:38_1_1"/><b>then </b><i><font color="Green">E90</font></i>: 
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 
<b>by </b><i>, <a class="ref" href="waybel_0.html#D20">WAYBEL_0:def 20</a></i>;<br/>
<a NAME="E16:38_1_1"/>
<font color="Maroon">g</font> is <a href="waybel_0.html#V22">directed-sups-preserving</a>
 
<b>by </b><i>, , <a class="ref" href="waybel21.html#T6">WAYBEL21:6</a></i>;<br/>
<a NAME="E17:38_1_1"/><b>then </b>
( <font color="Maroon">g</font> <a href="waybel_0.html#R4">preserves_sup_of</a> <font color="Maroon">D</font> &amp;  <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">D</font>,<font color="Maroon">Y</font> )
 
<b>by </b><i><a class="ref" href="waybel_0.html#D37">WAYBEL_0:def 37</a>, <a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/>
<a NAME="E18:38_1_1"/><b>then </b><i><font color="Green">E92</font></i>: 
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">D</font></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel_0.html#D31">WAYBEL_0:def 31</a></i>;<br/>
<a NAME="E19:38_1_1"/>
( <font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">D</font> is <a href="waybel_0.html#V1">directed</a> &amp; not <font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">D</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">V</font> is <a href="waybel11.html#V1">inaccessible_by_directed_joins</a> )
 
<b>by </b><i>, <a class="ref" href="waybel11.html#D4">WAYBEL11:def 4</a>, <a class="ref" href="yellow_2.html#T17">YELLOW_2:17</a></i>;<br/>
<a NAME="E20:38_1_1"/><b>then </b>
<font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">D</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">V</font>
 
<b>by </b><i>, , <a class="ref" href="waybel11.html#D1">WAYBEL11:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E22:38_1_1"/><i><font color="Green">E93</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">D</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E24:38_1_1"/><i><font color="Green">E109</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i>, <a class="ref" href="funct_2.html#T115">FUNCT_2:115</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font> <b>by </b><i/>;<br/>
<a NAME="E26:38_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">g</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">Y</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a>, <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>;<br/>
<a NAME="E27:38_1_1"/><b>then </b>
( <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">d</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">d</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">V</font> )
 
<b>by </b><i>, , <a class="ref" href="funct_2.html#T43">FUNCT_2:43</a>, <a class="ref" href="yellow_2.html#T10">YELLOW_2:10</a></i>;<br/>
<a NAME="E28:38_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K4">uparrow</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">V</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="waybel_0.html#D16">WAYBEL_0:def 16</a></i>;<br/>
<b>hence </b><a NAME="E29:38_1_1"/>
<font color="Maroon">D</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="waybel_0.html#K4">uparrow</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">V</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>


</div><b>end;</b></div><a NAME="E6:38_1"/><b>then </b>
 <a href="waybel_0.html#K4">uparrow</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">V</font></span>)</span> is  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="waybel11.html#D4">WAYBEL11:def 4</a></i>;<br/><b>hence </b><a NAME="E7:38_1"/>
 <a href="waybel_0.html#K4">uparrow</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel34.html#K1" target="_self">LowerAdj</a> <font color="Maroon">g</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">V</font></span>)</span> is  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font>
 <b>by </b><i>, <a class="ref" href="waybel_0.html#T13">WAYBEL_0:13</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:38"/><i><font color="Green">E110</font></i>: 
for <font color="Olive">X</font> being <a href="waybel11.html#V4">Scott</a>  <a href="yellow_9.html#M1">TopAugmentation</a> of <font color="Maroon">T</font><br/> for <font color="Olive">Y</font> being <a href="waybel11.html#V4">Scott</a>  <a href="yellow_9.html#M1">TopAugmentation</a> of <font color="Maroon">S</font><br/> for <font color="Olive">V</font> being <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Olive">X</font> holds   <a href="waybel_0.html#K4">uparrow</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel34.html#K1" target="_self">LowerAdj</a> <font color="Maroon">g</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Olive">V</font></span>)</span> is  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Olive">Y</font>
 ;<br/>

<b>consider </b><font color="Maroon">X</font> being  <a href="waybel11.html#V4">Scott</a>  <a href="yellow_9.html#M1">TopAugmentation</a> of <font color="Maroon">T</font>, <font color="Maroon">Y</font> being  <a href="waybel11.html#V4">Scott</a>  <a href="yellow_9.html#M1">TopAugmentation</a> of <font color="Maroon">S</font>;<br/>
<a NAME="E4:38"/><i><font color="Green">E111</font></i>: 
(  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">X</font> #) <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">T</font> #) &amp;  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">Y</font> #) <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</font> #) )
 
<b>by </b><i><a class="ref" href="yellow_9.html#D4">YELLOW_9:def 4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">g'</font> = <font color="Maroon">g</font> as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">Y</font>,<font color="Maroon">X</font> ;<br/>
<b>reconsider </b><font color="Maroon">g'</font> = <font color="Maroon">g'</font> as  <a href="waybel_0.html#V17">infs-preserving</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">Y</font>,<font color="Maroon">X</font> <b>by </b><i>, <a class="ref" href="waybel21.html#T6">WAYBEL21:6</a></i>;<br/>
<b>set </b><font color="Maroon">d</font> =  <a href="waybel34.html#K1" target="_self">LowerAdj</a> <font color="Maroon">g</font>;<br/>
<b>reconsider </b><font color="Maroon">d'</font> =  <a href="waybel34.html#K1" target="_self">LowerAdj</a> <font color="Maroon">g</font> as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font> <b>by </b><i/>;<br/>
<b>let </b><font color="Maroon">D</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D37">WAYBEL_0:def 37</a><br/>

<b>assume </b><a NAME="E8:38"/><i><font color="Green">E112</font></i>: 
( not <font color="Maroon">D</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">D</font> is <a href="waybel_0.html#V1">directed</a> )
 ;<br/>

<b>assume </b><a NAME="E9:38"/>
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">D</font>,<font color="Maroon">S</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D31">WAYBEL_0:def 31</a><br/>

<b>thus </b><a NAME="E10:38"/>
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">D</font>,<font color="Maroon">T</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/>

<a NAME="E11:38"/><i><font color="Green">E113</font></i>: 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">D</font></span>)</span> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D</font></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel17.html#T15">WAYBEL17:15</a></i>;<br/>
<b>reconsider </b><font color="Maroon">D'</font> = <font color="Maroon">D</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">D'</font> = <font color="Maroon">D'</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V1">directed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font> <b>by </b><i>, , <a class="ref" href="waybel_0.html#T3">WAYBEL_0:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">s</font> =  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font> <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">U'</font> = <span class="p1">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <span class="p2">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p3">(<span class="default"><font color="Maroon">g'</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">D'</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> ;<br/>
<a NAME="E15:38"/><i><font color="Green">E114</font></i>: 
<span class="p1">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <span class="p2">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p3">(<span class="default"><font color="Maroon">g'</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">D'</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a>  is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="waybel11.html#T12">WAYBEL11:12</a></i>;<br/>
<a NAME="E16:38"/><b>then </b>
 <a href="waybel_0.html#K4">uparrow</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel34.html#K1" target="_self">LowerAdj</a> <font color="Maroon">g</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <span class="p4">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p5">(<span class="default"><font color="Maroon">g'</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">D'</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> is  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font>
 
<b>by </b><i/>;<br/>
<a NAME="E17:38"/><b>then </b><i><font color="Green">E116</font></i>: 
 <a href="waybel_0.html#K4">uparrow</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel34.html#K1" target="_self">LowerAdj</a> <font color="Maroon">g</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <span class="p4">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p5">(<span class="default"><font color="Maroon">g'</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">D'</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> is  <a href="waybel_0.html#V13">upper</a> <a href="waybel11.html#V1">inaccessible</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font>
 
<b>by </b><i><a class="ref" href="waybel11.html#D4">WAYBEL11:def 4</a></i>;<br/>
<a NAME="E18:38"/><i><font color="Green">E117</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">D</font>,<font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/>
<a NAME="E19:38"/><b>then </b><i><font color="Green">E118</font></i>: 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">g'</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">D'</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">D</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="yellow_0.html#T26">YELLOW_0:26</a></i>;<br/>
<a NAME="E20:38"/>
 <a href="waybel_0.html#K5">downarrow</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p2">(<span class="default"><font color="Maroon">g'</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">D'</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K5">downarrow</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">D</font></span>)</span></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="waybel_0.html#T13">WAYBEL_0:13</a></i>;<br/>
<a NAME="E21:38"/><b>then </b><i><font color="Green">E119</font></i>: 
<span class="p1">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <span class="p2">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p3">(<span class="default"><font color="Maroon">g'</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">D'</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <span class="p2">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">D</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E22:38"/><i><font color="Green">E120</font></i>: 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">D</font></span>)</span> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">D</font></span>)</span>
 
;<br/>
<a NAME="E23:38"/>
<span class="p1"><a href="waybel_1.html#K1">[</a><span class="default"><font color="Maroon">g</font>,<span class="p2">(<span class="default"><a href="waybel34.html#K1" target="_self">LowerAdj</a> <font color="Maroon">g</font></span>)</span></span><a href="waybel_1.html#K1">]</a></span> is <a href="waybel_1.html#V3">Galois</a>
 
<b>by </b><i/>;<br/>
<a NAME="E24:38"/><b>then </b><i><font color="Green">E121</font></i>: 
( <span class="p1">(<span class="default"><a href="waybel34.html#K1" target="_self">LowerAdj</a> <font color="Maroon">g</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">g</font> <a href="yellow_2.html#R1">&lt;=</a>  <a href="grcat_1.html#K7">id</a> <font color="Maroon">S</font> &amp;  <a href="grcat_1.html#K7">id</a> <font color="Maroon">T</font> <a href="yellow_2.html#R1">&lt;=</a> <font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="waybel34.html#K1" target="_self">LowerAdj</a> <font color="Maroon">g</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="waybel_1.html#T19">WAYBEL_1:19</a></i>;<br/>
<a NAME="E25:38"/>
( <span class="p1">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">S</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel34.html#K1" target="_self">LowerAdj</a> <font color="Maroon">g</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">g</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel34.html#K1" target="_self">LowerAdj</a> <font color="Maroon">g</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a>, <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>;<br/>
<a NAME="E26:38"/><b>then </b>
<span class="p1">(<span class="default"><a href="waybel34.html#K1" target="_self">LowerAdj</a> <font color="Maroon">g</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D</font>
 
<b>by </b><i>, <a class="ref" href="yellow_2.html#T10">YELLOW_2:10</a></i>;<br/>
<a NAME="E27:38"/><b>then </b><i><font color="Green">E122</font></i>: 
<font color="Maroon">d'</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">g'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font></span>)</span> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">s</font>
 
<b>by </b><i>, <a class="ref" href="yellow_0.html#T1">YELLOW_0:1</a></i>;<br/>
<a NAME="E28:38"/>
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">D</font>,<font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/>
<a NAME="E29:38"/><b>then </b><i><font color="Green">E123</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D'</font>
 
<b>by </b><i>, <a class="ref" href="yellow_0.html#T26">YELLOW_0:26</a></i>;<br/>
<a NAME="E30:38"/>
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D</font></span>)</span> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">D</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:38_2"/>
not <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D</font></span>)</span> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">D</font></span>)</span>
 ;<br/>

<a NAME="E2:38_2"/><b>then </b>
( not <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K5">downarrow</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">D</font></span>)</span></span>)</span> &amp;  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">D</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K5">downarrow</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">D</font></span>)</span></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="waybel_0.html#T17">WAYBEL_0:17</a></i>;<br/>
<a NAME="E3:38_2"/><b>then </b><i><font color="Green">E124</font></i>: 
( <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D</font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <span class="p2">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p3">(<span class="default"><font color="Maroon">g'</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">D'</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a>  &amp; not  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">D</font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <span class="p2">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p3">(<span class="default"><font color="Maroon">g'</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">D'</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a>  )
 
<b>by </b><i>, <a class="ref" href="yellow_6.html#T10">YELLOW_6:10</a></i>;<br/>
<a NAME="E4:38_2"/><b>then </b>
( <font color="Maroon">d'</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">g'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">d'</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <span class="p3">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p4">(<span class="default"><font color="Maroon">g'</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">D'</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> &amp; <font color="Maroon">d'</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <span class="p3">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p4">(<span class="default"><font color="Maroon">g'</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">D'</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K4">uparrow</a> <span class="p1">(<span class="default"><font color="Maroon">d'</font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <span class="p4">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p5">(<span class="default"><font color="Maroon">g'</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">D'</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="funct_2.html#T43">FUNCT_2:43</a>, <a class="ref" href="waybel_0.html#T16">WAYBEL_0:16</a></i>;<br/>
<a NAME="E5:38_2"/><b>then </b>
( <font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K4">uparrow</a> <span class="p1">(<span class="default"><font color="Maroon">d'</font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <span class="p4">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p5">(<span class="default"><font color="Maroon">g'</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">D'</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> &amp;  <a href="waybel_0.html#K4">uparrow</a> <span class="p1">(<span class="default"><font color="Maroon">d'</font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <span class="p4">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p5">(<span class="default"><font color="Maroon">g'</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">D'</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K4">uparrow</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel34.html#K1" target="_self">LowerAdj</a> <font color="Maroon">g</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <span class="p4">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p5">(<span class="default"><font color="Maroon">g'</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">D'</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> )
 
<b>by </b><i>, , <a class="ref" href="waybel_0.html#T13">WAYBEL_0:13</a>, <a class="ref" href="waybel_0.html#D20">WAYBEL_0:def 20</a></i>;<br/>
<a NAME="E6:38_2"/><b>then </b>
<font color="Maroon">D'</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="waybel_0.html#K4">uparrow</a> <span class="p1">(<span class="default"><font color="Maroon">d'</font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <span class="p4">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p5">(<span class="default"><font color="Maroon">g'</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">D'</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="waybel11.html#D1">WAYBEL11:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:38_2"/><i><font color="Green">E125</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D'</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K4">uparrow</a> <span class="p1">(<span class="default"><font color="Maroon">d'</font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <span class="p4">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p5">(<span class="default"><font color="Maroon">g'</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">D'</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font> <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">u'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E11:38_2"/><i><font color="Green">E126</font></i>: 
( <font color="Maroon">u'</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">u'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">d'</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <span class="p3">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p4">(<span class="default"><font color="Maroon">g'</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">D'</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> )
 <b>by </b><i>, <a class="ref" href="waybel_0.html#D16">WAYBEL_0:def 16</a></i>;<br/>
<b>consider </b><font color="Maroon">u</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E13:38_2"/><i><font color="Green">E127</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <span class="p2">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p3">(<span class="default"><font color="Maroon">g'</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">D'</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">u'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font> )
 <b>by </b><i>, <a class="ref" href="funct_2.html#T115">FUNCT_2:115</a></i>;<br/>
<b>reconsider </b><font color="Maroon">u</font> = <font color="Maroon">u</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">u</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> <b>by </b><i/>;<br/>
<a NAME="E16:38_2"/>
<span class="p1">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">T</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>;<br/>
<a NAME="E17:38_2"/><b>then </b>
<font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="waybel34.html#K1" target="_self">LowerAdj</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font>
 
<b>by </b><i>, <a class="ref" href="yellow_2.html#T10">YELLOW_2:10</a></i>;<br/>
<a NAME="E18:38_2"/><b>then </b>
<font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel34.html#K1" target="_self">LowerAdj</a> <font color="Maroon">g</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>;<br/>
<a NAME="E19:38_2"/><b>then </b>
( <font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">g'</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">d'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u</font></span>)</span> &amp; <font color="Maroon">g'</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">d'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u</font></span>)</span> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">g'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> )
 
<b>by </b><i>, , , <a class="ref" href="orders_3.html#D5">ORDERS_3:def 5</a>, <a class="ref" href="yellow_0.html#T1">YELLOW_0:1</a></i>;<br/>
<a NAME="E20:38_2"/><b>then </b><i><font color="Green">E128</font></i>: 
<font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">g'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/>
<a NAME="E21:38_2"/>
<font color="Maroon">g'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">g'</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">D'</font>
 
<b>by </b><i>, <a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/>
<a NAME="E22:38_2"/><b>then </b>
<font color="Maroon">g'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">g'</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">D'</font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_2.html#T24">YELLOW_2:24</a></i>;<br/>
<a NAME="E23:38_2"/><b>then </b>
( <font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">g'</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">D'</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <span class="p2">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p3">(<span class="default"><font color="Maroon">g'</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">D'</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a>  is <a href="waybel_0.html#V13">upper</a> )
 
<b>by </b><i>, , <a class="ref" href="orders_2.html#T26">ORDERS_2:26</a>, <a class="ref" href="waybel11.html#D4">WAYBEL11:def 4</a></i>;<br/>
<a NAME="E24:38_2"/><b>then </b>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">g'</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">D'</font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <span class="p2">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p3">(<span class="default"><font color="Maroon">g'</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">D'</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i>, <a class="ref" href="waybel_0.html#D20">WAYBEL_0:def 20</a></i>;<br/>
<b>hence </b><a NAME="E25:38_2"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="waybel34.html#T1" target="_self">Th1</a>, <a class="ref" href="yellow_0.html#T26">YELLOW_0:26</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E31:38"/>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">D</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="orders_2.html#T25">ORDERS_2:25</a></i>;<br/>


</div>
