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

<span class="kw">let </span><font color="Maroon" title="c1">S</font>, <font color="Maroon" title="c2">T</font> be   <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a>  <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">g</font> being  <a href="waybel_0.html#V17" title="WAYBEL_0:attr.17">infs-preserving</a>  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font>  st <font color="Olive" title="b1">g</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  holds <br/>for <font color="Olive" title="b2">X</font> being  <a href="waybel11.html#V4" title="WAYBEL11:attr.4">Scott</a>   <a href="yellow_9.html#M1" title="YELLOW_9:mode.1">TopAugmentation</a> of <font color="Maroon" title="c2">T</font><br/> for <font color="Olive" title="b3">Y</font> being  <a href="waybel11.html#V4" title="WAYBEL11:attr.4">Scott</a>   <a href="yellow_9.html#M1" title="YELLOW_9:mode.1">TopAugmentation</a> of <font color="Maroon" title="c1">S</font><br/> for <font color="Olive" title="b4">d</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Olive" title="b2">X</font>,<font color="Olive" title="b3">Y</font>  st <font color="Olive" title="b4">d</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="waybel34.html#K1" title="WAYBEL34:func.1">LowerAdj</a> <font color="Olive" title="b1">g</font> holds <br/>( <font color="Olive" title="b1">g</font> is  <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a>  iff <font color="Olive" title="b4">d</font> is  <a href="t_0topsp.html#V1" title="T_0TOPSP:attr.1">open</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">g</font> be   <a href="waybel_0.html#V17" title="WAYBEL_0:attr.17">infs-preserving</a>  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">g</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  implies for <font color="Olive" title="b1">X</font> being  <a href="waybel11.html#V4" title="WAYBEL11:attr.4">Scott</a>   <a href="yellow_9.html#M1" title="YELLOW_9:mode.1">TopAugmentation</a> of <font color="Maroon" title="c2">T</font><br/> for <font color="Olive" title="b2">Y</font> being  <a href="waybel11.html#V4" title="WAYBEL11:attr.4">Scott</a>   <a href="yellow_9.html#M1" title="YELLOW_9:mode.1">TopAugmentation</a> of <font color="Maroon" title="c1">S</font><br/> for <font color="Olive" title="b3">d</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Olive" title="b1">X</font>,<font color="Olive" title="b2">Y</font>  st <font color="Olive" title="b3">d</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="waybel34.html#K1" title="WAYBEL34:func.1">LowerAdj</a> <font color="Maroon" title="c3">g</font> holds <br/>( <font color="Maroon" title="c3">g</font> is  <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a>  iff <font color="Olive" title="b3">d</font> is  <a href="t_0topsp.html#V1" title="T_0TOPSP:attr.1">open</a>  ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:52"/><span class="lab"><font color="Green" title="E33">A1</font></span>: 
<font color="Maroon" title="c3">g</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">X</font> being  <a href="waybel11.html#V4" title="WAYBEL11:attr.4">Scott</a>   <a href="yellow_9.html#M1" title="YELLOW_9:mode.1">TopAugmentation</a> of <font color="Maroon" title="c2">T</font><br/> for <font color="Olive" title="b2">Y</font> being  <a href="waybel11.html#V4" title="WAYBEL11:attr.4">Scott</a>   <a href="yellow_9.html#M1" title="YELLOW_9:mode.1">TopAugmentation</a> of <font color="Maroon" title="c1">S</font><br/> for <font color="Olive" title="b3">d</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Olive" title="b1">X</font>,<font color="Olive" title="b2">Y</font>  st <font color="Olive" title="b3">d</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="waybel34.html#K1" title="WAYBEL34:func.1">LowerAdj</a> <font color="Maroon" title="c3">g</font> holds <br/>( <font color="Maroon" title="c3">g</font> is  <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a>  iff <font color="Olive" title="b3">d</font> is  <a href="t_0topsp.html#V1" title="T_0TOPSP:attr.1">open</a>  )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c4">X</font> be   <a href="waybel11.html#V4" title="WAYBEL11:attr.4">Scott</a>   <a href="yellow_9.html#M1" title="YELLOW_9:mode.1">TopAugmentation</a> of <font color="Maroon" title="c2">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">Y</font> being  <a href="waybel11.html#V4" title="WAYBEL11:attr.4">Scott</a>   <a href="yellow_9.html#M1" title="YELLOW_9:mode.1">TopAugmentation</a> of <font color="Maroon" title="c1">S</font><br/> for <font color="Olive" title="b2">d</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c4">X</font>,<font color="Olive" title="b1">Y</font>  st <font color="Olive" title="b2">d</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="waybel34.html#K1" title="WAYBEL34:func.1">LowerAdj</a> <font color="Maroon" title="c3">g</font> holds <br/>( <font color="Maroon" title="c3">g</font> is  <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a>  iff <font color="Olive" title="b2">d</font> is  <a href="t_0topsp.html#V1" title="T_0TOPSP:attr.1">open</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">Y</font> be   <a href="waybel11.html#V4" title="WAYBEL11:attr.4">Scott</a>   <a href="yellow_9.html#M1" title="YELLOW_9:mode.1">TopAugmentation</a> of <font color="Maroon" title="c1">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">d</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c5">Y</font>  st <font color="Olive" title="b1">d</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="waybel34.html#K1" title="WAYBEL34:func.1">LowerAdj</a> <font color="Maroon" title="c3">g</font> holds <br/>( <font color="Maroon" title="c3">g</font> is  <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a>  iff <font color="Olive" title="b1">d</font> is  <a href="t_0topsp.html#V1" title="T_0TOPSP:attr.1">open</a>  )</span><br/>



<a NAME="E2:52"/>
<span class="p1"><a href="waybel_1.html#K1" title="WAYBEL_1:func.1">[</a><span class="default"><font color="Maroon" title="c3">g</font>,<span class="p2">(<span class="default"><a href="waybel34.html#K1" title="WAYBEL34:func.1">LowerAdj</a> <font color="Maroon" title="c3">g</font></span>)</span></span><a href="waybel_1.html#K1" title="WAYBEL_1:func.1">]</a></span> is  <a href="waybel_1.html#V3" title="WAYBEL_1:attr.3">Galois</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel34.html#D1" target="_self" title="WAYBEL34:def.1">Def1</a></span>;<br/>
<a NAME="E3:52"/><span class="kw">then </span>
 <a href="waybel34.html#K1" title="WAYBEL34:func.1">LowerAdj</a> <font color="Maroon" title="c3">g</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:52"><span class="lab"><font color="Green" title="E33">A1</font></span></a>, <a class="ref" href="waybel_1.html#T29" title="WAYBEL_1:th.29">WAYBEL_1:29</a></span>;<br/>
<a NAME="E4:52"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A2</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="waybel34.html#K1" title="WAYBEL34:func.1">LowerAdj</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></span>;<br/>
<a NAME="E5:52"/><span class="lab"><font color="Green" title="E35">A3</font></span>: 
 <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">Y</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c5">Y</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">S</font> #)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_9.html#D4" title="YELLOW_9:def.4">YELLOW_9:def 4</a></span>;<br/>
<a NAME="E6:52"/><span class="lab"><font color="Green" title="E36">A4</font></span>: 
 <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">X</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c4">X</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c2">T</font> #)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_9.html#D4" title="YELLOW_9:def.4">YELLOW_9:def 4</a></span>;<br/>
<a NAME="E7:52"/><span class="lab"><font color="Green" title="E37">A5</font></span>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c5">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">Y</font>
 
;<br/>
<span class="kw">let </span><font color="Maroon" title="c6">d</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c5">Y</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">d</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="waybel34.html#K1" title="WAYBEL34:func.1">LowerAdj</a> <font color="Maroon" title="c3">g</font> implies ( <font color="Maroon" title="c3">g</font> is  <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a>  iff <font color="Maroon" title="c6">d</font> is  <a href="t_0topsp.html#V1" title="T_0TOPSP:attr.1">open</a>  ) )</span><br/>

<span class="kw">assume </span><a NAME="E8:52"/><span class="lab"><font color="Green" title="E38">A6</font></span>: 
<font color="Maroon" title="c6">d</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="waybel34.html#K1" title="WAYBEL34:func.1">LowerAdj</a> <font color="Maroon" title="c3">g</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">g</font> is  <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a>  iff <font color="Maroon" title="c6">d</font> is  <a href="t_0topsp.html#V1" title="T_0TOPSP:attr.1">open</a>  )</span><br/>

<a NAME="E9:52"/><span class="lab"><font color="Green" title="E39">A7</font></span>: 
<font color="Maroon" title="c5">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">d</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">Y</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c5">Y</font> #)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:52"><span class="lab"><font color="Green" title="E34">A2</font></span></a>, <a class="txt" href="#E5:52"><span class="lab"><font color="Green" title="E35">A3</font></span></a>, <a class="txt" href="#E7:52"><span class="lab"><font color="Green" title="E37">A5</font></span></a>, <a class="txt" href="#E8:52"><span class="lab"><font color="Green" title="E38">A6</font></span></a>, <a class="ref" href="tsep_1.html#T100" title="TSEP_1:th.100">TSEP_1:100</a></span>;<br/>
<span class="kw">thus </span><a NAME="E10:52"/>
( <font color="Maroon" title="c3">g</font> is  <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a>  implies <font color="Maroon" title="c6">d</font> is  <a href="t_0topsp.html#V1" title="T_0TOPSP:attr.1">open</a>  )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">d</font> is  <a href="t_0topsp.html#V1" title="T_0TOPSP:attr.1">open</a>  implies <font color="Maroon" title="c3">g</font> is  <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:52_1"/>
<font color="Maroon" title="c3">g</font> is  <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">d</font> is  <a href="t_0topsp.html#V1" title="T_0TOPSP:attr.1">open</a> </span><br/>

<a NAME="E2:52_1"/><span class="kw">then </span>
for <font color="Olive" title="b1">V</font> being  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c4">X</font> holds   <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel34.html#K1" title="WAYBEL34:func.1">LowerAdj</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">V</font></span>)</span> is   <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c5">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel34.html#T23" target="_self" title="WAYBEL34:th.23">Th23</a></span>;<br/>
<a NAME="E3:52_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A8</font></span>: 
<font color="Maroon" title="c6">d</font> is  <a href="waybel34.html#V2" title="WAYBEL34:attr.2">relatively_open</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:52"><span class="lab"><font color="Green" title="E38">A6</font></span></a>, <a class="ref" href="waybel34.html#T28" target="_self" title="WAYBEL34:th.28">Th28</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c7">V</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c4">X</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="t_0topsp.html#D2" title="T_0TOPSP:def.2">T_0TOPSP:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c7">V</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  or <font color="Maroon" title="c6">d</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c7">V</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E4:52_1"/>
<font color="Maroon" title="c7">V</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">d</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c7">V</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> </span><br/>

<a NAME="E5:52_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">d</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c7">V</font> is   <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c5">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">d</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:52_1"><span class="lab"><font color="Green" title="E40">A8</font></span></a>, <a class="ref" href="waybel34.html#D9" target="_self" title="WAYBEL34:def.9">Def9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:52_1"/>
<font color="Maroon" title="c6">d</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c7">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c5">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:52"><span class="lab"><font color="Green" title="E39">A7</font></span></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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

<span class="kw">assume </span><a NAME="E11:52"/><span class="lab"><font color="Green" title="E40">A9</font></span>: 
for <font color="Olive" title="b1">V</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c4">X</font>  st <font color="Olive" title="b1">V</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  holds <br/><font color="Maroon" title="c6">d</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">V</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="t_0topsp.html#D2" title="T_0TOPSP:def.2">T_0TOPSP:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">g</font> is  <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a> </span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">X9</font> be   <a href="waybel11.html#V4" title="WAYBEL11:attr.4">Scott</a>   <a href="yellow_9.html#M1" title="YELLOW_9:mode.1">TopAugmentation</a> of <font color="Maroon" title="c2">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">Y9</font> being  <a href="waybel11.html#V4" title="WAYBEL11:attr.4">Scott</a>   <a href="yellow_9.html#M1" title="YELLOW_9:mode.1">TopAugmentation</a> of <font color="Maroon" title="c1">S</font><br/> for <font color="Olive" title="b2">V9</font> being  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c7">X9</font> holds   <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel34.html#K1" title="WAYBEL34:func.1">LowerAdj</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b2">V9</font></span>)</span> is   <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Olive" title="b1">Y9</font></span><br/><span class="kw">let </span><font color="Maroon" title="c8">Y9</font> be   <a href="waybel11.html#V4" title="WAYBEL11:attr.4">Scott</a>   <a href="yellow_9.html#M1" title="YELLOW_9:mode.1">TopAugmentation</a> of <font color="Maroon" title="c1">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">V9</font> being  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c7">X9</font> holds   <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel34.html#K1" title="WAYBEL34:func.1">LowerAdj</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">V9</font></span>)</span> is   <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c8">Y9</font></span><br/><span class="kw">let </span><font color="Maroon" title="c9">V9</font> be   <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c7">X9</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel34.html#K1" title="WAYBEL34:func.1">LowerAdj</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c9">V9</font></span>)</span> is   <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c8">Y9</font></span><br/><a NAME="E1:52_2"/><span class="lab"><font color="Green" title="E41">A10</font></span>: 
 <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">X9</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c7">X9</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c2">T</font> #)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_9.html#D4" title="YELLOW_9:def.4">YELLOW_9:def 4</a></span>;<br/><a NAME="E2:52_2"/><span class="lab"><font color="Green" title="E42">A11</font></span>: 
 <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y9</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c8">Y9</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">S</font> #)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_9.html#D4" title="YELLOW_9:def.4">YELLOW_9:def 4</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c10">V</font> = <font color="Maroon" title="c9">V9</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c4">X</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:52"><span class="lab"><font color="Green" title="E36">A4</font></span></a>, <a class="txt" href="#E1:52_2"><span class="lab"><font color="Green" title="E41">A10</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c11">V</font> = <font color="Maroon" title="c10">V</font> as   <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c4">X</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:52"><span class="lab"><font color="Green" title="E36">A4</font></span></a>, <a class="txt" href="#E1:52_2"><span class="lab"><font color="Green" title="E41">A10</font></span></a>, <a class="ref" href="yellow_9.html#T50" title="YELLOW_9:th.50">YELLOW_9:50</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c12">d9</font> = <font color="Maroon" title="c6">d</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c7">X9</font>,<font color="Maroon" title="c8">Y9</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:52"><span class="lab"><font color="Green" title="E35">A3</font></span></a>, <a class="txt" href="#E6:52"><span class="lab"><font color="Green" title="E36">A4</font></span></a>, <a class="txt" href="#E1:52_2"><span class="lab"><font color="Green" title="E41">A10</font></span></a>, <a class="txt" href="#E2:52_2"><span class="lab"><font color="Green" title="E42">A11</font></span></a></span>;<br/><a NAME="E6:52_2"/>
<font color="Maroon" title="c6">d</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c11">V</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:52"><span class="lab"><font color="Green" title="E40">A9</font></span></a></span>;<br/><a NAME="E7:52_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A12</font></span>: 
<font color="Maroon" title="c12">d9</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c9">V9</font> is   <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c8">Y9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:52"><span class="lab"><font color="Green" title="E35">A3</font></span></a>, <a class="txt" href="#E2:52_2"><span class="lab"><font color="Green" title="E42">A11</font></span></a>, <a class="ref" href="yellow_9.html#T50" title="YELLOW_9:th.50">YELLOW_9:50</a></span>;<br/><a NAME="E8:52_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">d9</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c9">V9</font> is  <a href="waybel_0.html#V13" title="WAYBEL_0:attr.13">upper</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel11.html#D4" title="WAYBEL11:def.4">WAYBEL11:def 4</a></span>;<br/><a NAME="E9:52_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A13</font></span>: 
 <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">d9</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c9">V9</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">d9</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c9">V9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_0.html#T24" title="WAYBEL_0:th.24">WAYBEL_0:24</a></span>;<br/><a NAME="E10:52_2"/>
<font color="Maroon" title="c12">d9</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c9">V9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">d9</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c9">V9</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_0.html#T16" title="WAYBEL_0:th.16">WAYBEL_0:16</a></span>;<br/><a NAME="E11:52_2"/><span class="kw">then </span>
 <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">d9</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c9">V9</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">d9</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c9">V9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:52_2"><span class="lab"><font color="Green" title="E44">A13</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/><span class="kw">hence </span><a NAME="E12:52_2"/>
 <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel34.html#K1" title="WAYBEL34:func.1">LowerAdj</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c9">V9</font></span>)</span> is   <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c8">Y9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:52"><span class="lab"><font color="Green" title="E38">A6</font></span></a>, <a class="txt" href="#E2:52_2"><span class="lab"><font color="Green" title="E42">A11</font></span></a>, <a class="txt" href="#E7:52_2"><span class="lab"><font color="Green" title="E43">A12</font></span></a>, <a class="ref" href="waybel_0.html#T13" title="WAYBEL_0:th.13">WAYBEL_0:13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E13:52"/>
<font color="Maroon" title="c3">g</font> is  <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel34.html#T23" target="_self" title="WAYBEL34:th.23">Th23</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
