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

<b>let </b><font color="Maroon" title="c1">S</font>, <font color="Maroon" title="c2">S'</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ;<br/><b>let </b><font color="Maroon" title="c3">T</font>, <font color="Maroon" title="c4">T'</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="orders_2.html#V2" title="ORDERS_2:attr.2">reflexive</a> <a href="orders_2.html#V4" title="ORDERS_2:attr.4">antisymmetric</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ;<br/>



<b>assume </b><a NAME="E1:41"/><i><font color="Green" title="E28">A1</font></i>: 
(  <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> #) <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">S'</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c2">S'</font> #) &amp;  <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="c3">T</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c3">T</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="c4">T'</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c4">T'</font> #) )
 ;<br/>

<a NAME="E2:41"/><i><font color="Green" title="E29">A2</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel27.html#K2" title="WAYBEL27:func.2">UPS</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c3">T</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 <span class="p1">(<span class="default"><a href="waybel27.html#K2" title="WAYBEL27:func.2">UPS</a> <font color="Maroon" title="c2">S'</font>,<font color="Maroon" title="c4">T'</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:41_1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel27.html#K2" title="WAYBEL27:func.2">UPS</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c3">T</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel27.html#K2" title="WAYBEL27:func.2">UPS</a> <font color="Maroon" title="c2">S'</font>,<font color="Maroon" title="c4">T'</font></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:41_1_1"/>
<font color="Maroon" title="c5">x</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 <span class="p1">(<span class="default"><a href="waybel27.html#K2" title="WAYBEL27:func.2">UPS</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c3">T</font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c6">x1</font> = <font color="Maroon" title="c5">x</font> as  <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-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="c3">T</font> <b>by </b><i><a class="ref" href="waybel27.html#D4" target="_self" title="WAYBEL27:def.4">Def4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">y</font> = <font color="Maroon" title="c6">x1</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">S'</font>,<font color="Maroon" title="c4">T'</font> <b>by </b><i><a class="txt" href="waybel27.html#E1:41"><i><font color="Green" title="E28">A1</font></i></a></i>;<br/>
<a NAME="E4:41_1_1"/>
<font color="Maroon" title="c7">y</font> is <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">X</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">S'</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D37" title="WAYBEL_0:def.37">WAYBEL_0:def 37</a><br/>

<b>assume </b><a NAME="E1:41_1_1_1"/><i><font color="Green" title="E30">A3</font></i>: 
( not <font color="Maroon" title="c8">X</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; <font color="Maroon" title="c8">X</font> is <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> )
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c9">Y</font> = <font color="Maroon" title="c8">X</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">S</font> <b>by </b><i><a class="txt" href="waybel27.html#E1:41"><i><font color="Green" title="E28">A1</font></i></a></i>;<br/>
<a NAME="E3:41_1_1_1"/>
( not <font color="Maroon" title="c9">Y</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; <font color="Maroon" title="c9">Y</font> is <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> )
 
<b>by </b><i><a class="txt" href="waybel27.html#E1:41"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="waybel27.html#E1:41_1_1_1"><i><font color="Green" title="E30">A3</font></i></a>, <a class="ref" href="waybel_0.html#T3" title="WAYBEL_0:th.3">WAYBEL_0:3</a></i>;<br/>
<a NAME="E4:41_1_1_1"/><b>then </b>
<font color="Maroon" title="c6">x1</font> <a href="waybel_0.html#R4" title="WAYBEL_0:pred.4">preserves_sup_of</a> <font color="Maroon" title="c9">Y</font>
 
<b>by </b><i><a class="ref" href="waybel_0.html#D37" title="WAYBEL_0:def.37">WAYBEL_0:def 37</a></i>;<br/>
<b>hence </b><a NAME="E5:41_1_1_1"/>
<font color="Maroon" title="c7">y</font> <a href="waybel_0.html#R4" title="WAYBEL_0:pred.4">preserves_sup_of</a> <font color="Maroon" title="c8">X</font>
 <b>by </b><i><a class="txt" href="waybel27.html#E1:41"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="waybel_0.html#T65" title="WAYBEL_0:th.65">WAYBEL_0:65</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:41_1_1"/>
<font color="Maroon" title="c5">x</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 <span class="p1">(<span class="default"><a href="waybel27.html#K2" title="WAYBEL27:func.2">UPS</a> <font color="Maroon" title="c2">S'</font>,<font color="Maroon" title="c4">T'</font></span>)</span>
 <b>by </b><i><a class="ref" href="waybel27.html#D4" target="_self" title="WAYBEL27:def.4">Def4</a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:41_1"/>
<font color="Maroon" title="c5">x</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 <span class="p1">(<span class="default"><a href="waybel27.html#K2" title="WAYBEL27:func.2">UPS</a> <font color="Maroon" title="c2">S'</font>,<font color="Maroon" title="c4">T'</font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c6">x1</font> = <font color="Maroon" title="c5">x</font> as  <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">S'</font>,<font color="Maroon" title="c4">T'</font> <b>by </b><i><a class="ref" href="waybel27.html#D4" target="_self" title="WAYBEL27:def.4">Def4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">y</font> = <font color="Maroon" title="c6">x1</font> as   <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="c3">T</font> <b>by </b><i><a class="txt" href="waybel27.html#E1:41"><i><font color="Green" title="E28">A1</font></i></a></i>;<br/>
<a NAME="E5:41_1"/>
<font color="Maroon" title="c7">y</font> is <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">X</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">S</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D37" title="WAYBEL_0:def.37">WAYBEL_0:def 37</a><br/>

<b>assume </b><a NAME="E1:41_1_2"/><i><font color="Green" title="E30">A4</font></i>: 
( not <font color="Maroon" title="c8">X</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; <font color="Maroon" title="c8">X</font> is <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> )
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c9">Y</font> = <font color="Maroon" title="c8">X</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">S'</font> <b>by </b><i><a class="txt" href="waybel27.html#E1:41"><i><font color="Green" title="E28">A1</font></i></a></i>;<br/>
<a NAME="E3:41_1_2"/>
( not <font color="Maroon" title="c9">Y</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; <font color="Maroon" title="c9">Y</font> is <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> )
 
<b>by </b><i><a class="txt" href="waybel27.html#E1:41"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="waybel27.html#E1:41_1_2"><i><font color="Green" title="E30">A4</font></i></a>, <a class="ref" href="waybel_0.html#T3" title="WAYBEL_0:th.3">WAYBEL_0:3</a></i>;<br/>
<a NAME="E4:41_1_2"/><b>then </b>
<font color="Maroon" title="c6">x1</font> <a href="waybel_0.html#R4" title="WAYBEL_0:pred.4">preserves_sup_of</a> <font color="Maroon" title="c9">Y</font>
 
<b>by </b><i><a class="ref" href="waybel_0.html#D37" title="WAYBEL_0:def.37">WAYBEL_0:def 37</a></i>;<br/>
<b>hence </b><a NAME="E5:41_1_2"/>
<font color="Maroon" title="c7">y</font> <a href="waybel_0.html#R4" title="WAYBEL_0:pred.4">preserves_sup_of</a> <font color="Maroon" title="c8">X</font>
 <b>by </b><i><a class="txt" href="waybel27.html#E1:41"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="waybel_0.html#T65" title="WAYBEL_0:th.65">WAYBEL_0:65</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:41_1"/>
<font color="Maroon" title="c5">x</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 <span class="p1">(<span class="default"><a href="waybel27.html#K2" title="WAYBEL27:func.2">UPS</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c3">T</font></span>)</span>
 <b>by </b><i><a class="ref" href="waybel27.html#D4" target="_self" title="WAYBEL27:def.4">Def4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:41"/><i><font color="Green" title="E30">A5</font></i>: 
 <a href="waybel27.html#K2" title="WAYBEL27:func.2">UPS</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c3">T</font> is  <a href="yellow_0.html#V4" title="YELLOW_0:attr.4">full</a>  <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of <font color="Maroon" title="c3">T</font> <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>
 
<b>by </b><i><a class="ref" href="waybel27.html#D4" target="_self" title="WAYBEL27:def.4">Def4</a></i>;<br/>
<a NAME="E4:41"/>
<font color="Maroon" title="c3">T</font> <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">T'</font> <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S'</font>
 
<b>by </b><i><a class="txt" href="waybel27.html#E1:41"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="waybel27.html#T15" target="_self" title="WAYBEL27:th.15">Th15</a></i>;<br/>
<a NAME="E5:41"/><b>then </b>
 <a href="waybel27.html#K2" title="WAYBEL27:func.2">UPS</a> <font color="Maroon" title="c2">S'</font>,<font color="Maroon" title="c4">T'</font> is  <a href="yellow_0.html#V4" title="YELLOW_0:attr.4">full</a>  <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of <font color="Maroon" title="c3">T</font> <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>
 
<b>by </b><i><a class="ref" href="waybel27.html#D4" target="_self" title="WAYBEL27:def.4">Def4</a></i>;<br/>
<b>hence </b><a NAME="E6:41"/>
 <a href="waybel27.html#K2" title="WAYBEL27:func.2">UPS</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c3">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel27.html#K2" title="WAYBEL27:func.2">UPS</a> <font color="Maroon" title="c2">S'</font>,<font color="Maroon" title="c4">T'</font>
 <b>by </b><i><a class="txt" href="waybel27.html#E2:41"><i><font color="Green" title="E29">A2</font></i></a>, <a class="txt" href="waybel27.html#E3:41"><i><font color="Green" title="E30">A5</font></i></a>, <a class="ref" href="yellow_0.html#T58" title="YELLOW_0:th.58">YELLOW_0:58</a></i>;<br/>


</div>
