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

<span class="kw">let </span><font color="Maroon" title="c1">L</font> be  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="orders_2.html#NM1" title="ORDERS_2:NM.1">Poset</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">S</font> being non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="yellow_0.html#V4" title="YELLOW_0:attr.4">full</a>   <a href="waybel_0.html#V4" title="WAYBEL_0:attr.4">directed-sups-inheriting</a>   <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of <font color="Maroon" title="c1">L</font><br/> for <font color="Olive" title="b2">X</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds  <font color="Olive" title="b1">S</font> <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> <font color="Olive" title="b2">X</font> is   <a href="waybel_0.html#V4" title="WAYBEL_0:attr.4">directed-sups-inheriting</a>   <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of <font color="Maroon" title="c1">L</font> <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> <font color="Olive" title="b2">X</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">S</font> be  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="yellow_0.html#V4" title="YELLOW_0:attr.4">full</a>   <a href="waybel_0.html#V4" title="WAYBEL_0:attr.4">directed-sups-inheriting</a>   <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of <font color="Maroon" title="c1">L</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">X</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds  <font color="Maroon" title="c2">S</font> <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> <font color="Olive" title="b1">X</font> is   <a href="waybel_0.html#V4" title="WAYBEL_0:attr.4">directed-sups-inheriting</a>   <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of <font color="Maroon" title="c1">L</font> <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> <font color="Olive" title="b1">X</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">X</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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="c2">S</font> <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> <font color="Maroon" title="c3">X</font> is   <a href="waybel_0.html#V4" title="WAYBEL_0:attr.4">directed-sups-inheriting</a>   <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of <font color="Maroon" title="c1">L</font> <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> <font color="Maroon" title="c3">X</font></span><br/>





<span class="kw">reconsider </span><font color="Maroon" title="c4">SX</font> = <font color="Maroon" title="c2">S</font> <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> <font color="Maroon" title="c3">X</font> as  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <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="c1">L</font> <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> <font color="Maroon" title="c3">X</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow16.html#T41" target="_self" title="YELLOW16:th.41">Th41</a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> , non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="orders_2.html#V3" title="ORDERS_2:attr.3">reflexive</a>   <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ] <span class="kw">means </span>$1 is  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of $2;<br/>
<div><span class="lab"><font color="Green" title="E39">A1</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> <font color="Maroon" title="c3">X</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c2">S</font> <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> <font color="Maroon" title="c3">X</font>] implies for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">X</font> holds  <font color="Maroon">S<sub>1</sub></font>[ <a href="yellow16.html#K1" title="YELLOW16:func.1">pi</a> <font color="Maroon" title="c5">A</font>,<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c2">S</font>] )</span><br/><span class="kw">assume </span><a NAME="E1:63_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c2">S</font> <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> <font color="Maroon" title="c3">X</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">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">X</font> holds  <font color="Maroon">S<sub>1</sub></font>[ <a href="yellow16.html#K1" title="YELLOW16:func.1">pi</a> <font color="Maroon" title="c5">A</font>,<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c2">S</font>]</span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">B</font> = <font color="Maroon" title="c5">A</font> as  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</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="c2">S</font> <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> <font color="Maroon" title="c3">X</font></span>)</span> ;<br/><span class="kw">let </span><font color="Maroon" title="c7">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">X</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">S<sub>1</sub></font>[ <a href="yellow16.html#K1" title="YELLOW16:func.1">pi</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c2">S</font>]</span><br/><a NAME="E3:63_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="yellow16.html#K2" title="YELLOW16:func.2">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">S</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>;<br/><a NAME="E4:63_1"/><span class="kw">then </span>
 <a href="yellow16.html#K1" title="YELLOW16:func.1">pi</a> <font color="Maroon" title="c6">B</font>,<font color="Maroon" title="c7">i</font> is  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">S</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow16.html#T37" target="_self" title="YELLOW16:th.37">Th37</a></span>;<br/><span class="kw">hence </span><a NAME="E5:63_1"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="yellow16.html#K1" title="YELLOW16:func.1">pi</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c2">S</font>]
 ; <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>
<div><span class="lab"><font color="Green" title="E40">A2</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="63_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">X</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">S</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">S<sub>1</sub></font>[<font color="Maroon" title="c5">X</font>,<font color="Maroon" title="c2">S</font>] implies <font color="Maroon" title="c2">S</font> <a href="yellow16.html#R5" title="YELLOW16:pred.5">inherits_sup_of</a> <font color="Maroon" title="c5">X</font>,<font color="Maroon" title="c1">L</font> )</span><br/><span class="kw">assume </span><a NAME="E1:63_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">X</font>,<font color="Maroon" title="c2">S</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="c2">S</font> <a href="yellow16.html#R5" title="YELLOW16:pred.5">inherits_sup_of</a> <font color="Maroon" title="c5">X</font>,<font color="Maroon" title="c1">L</font></span><br/><a NAME="E2:63_2"/><span class="kw">then </span>
(  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c5">X</font>,<font color="Maroon" title="c1">L</font> implies (  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c5">X</font>,<font color="Maroon" title="c2">S</font> &amp;  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c5">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c5">X</font>,<font color="Maroon" title="c1">L</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_0.html#T7" title="WAYBEL_0:th.7">WAYBEL_0:7</a></span>;<br/><span class="kw">hence </span><a NAME="E3:63_2"/>
<font color="Maroon" title="c2">S</font> <a href="yellow16.html#R5" title="YELLOW16:pred.5">inherits_sup_of</a> <font color="Maroon" title="c5">X</font>,<font color="Maroon" title="c1">L</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow16.html#T42" target="_self" title="YELLOW16:th.42">Th42</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>
<a NAME="E4:63"/>
<font color="Maroon" title="c4">SX</font> is  <a href="waybel_0.html#V4" title="WAYBEL_0:attr.4">directed-sups-inheriting</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">A</font> be   <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c4">SX</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="waybel_0.html#D4" title="WAYBEL_0:def.4">WAYBEL_0:def 4</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="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or  not  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c1">L</font> <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> <font color="Maroon" title="c3">X</font> or  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c5">A</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">SX</font> )</span><br/>

<a NAME="E1:63_3"/>
for <font color="Olive" title="b1">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> <font color="Maroon" title="c3">X</font></span>)</span>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">A</font>,<font color="Maroon" title="c2">S</font> <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> <font color="Maroon" title="c3">X</font>] holds <br/><font color="Maroon" title="c2">S</font> <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> <font color="Maroon" title="c3">X</font> <a href="yellow16.html#R5" title="YELLOW16:pred.5">inherits_sup_of</a> <font color="Olive" title="b1">A</font>,<font color="Maroon" title="c1">L</font> <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> <font color="Maroon" title="c3">X</font>
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="yellow16.html#S2" target="_self" title="YELLOW16:sch.2">YELLOW16:sch 2</a>(<a class="txt" href="#E2:63"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="txt" href="#E3:63"><span class="lab"><font color="Green" title="E40">A2</font></span></a>);<br/></span>
<a NAME="E2:63_3"/><span class="kw">then </span>
( <font color="Maroon" title="c5">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies <font color="Maroon" title="c2">S</font> <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> <font color="Maroon" title="c3">X</font> <a href="yellow16.html#R5" title="YELLOW16:pred.5">inherits_sup_of</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c1">L</font> <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> <font color="Maroon" title="c3">X</font> )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:63_3"/>
( <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or  not  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c1">L</font> <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> <font color="Maroon" title="c3">X</font> or  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c5">A</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">SX</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow16.html#D6" target="_self" title="YELLOW16:def.6">Def6</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="E5:63"/>
<font color="Maroon" title="c2">S</font> <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> <font color="Maroon" title="c3">X</font> is   <a href="waybel_0.html#V4" title="WAYBEL_0:attr.4">directed-sups-inheriting</a>   <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of <font color="Maroon" title="c1">L</font> <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> <font color="Maroon" title="c3">X</font>
 ; <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>
