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

<span class="kw">let </span><font color="Maroon" title="c1">S1</font>, <font color="Maroon" title="c2">T1</font>, <font color="Maroon" title="c3">S2</font>, <font color="Maroon" title="c4">T2</font> be  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <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">S1</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">S1</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="c3">S2</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c3">S2</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="c2">T1</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c2">T1</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">T2</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c4">T2</font> #) implies for <font color="Olive" title="b1">f1</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">T1</font><br/> for <font color="Olive" title="b2">f2</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c3">S2</font>,<font color="Maroon" title="c4">T2</font>  st <font color="Olive" title="b1">f1</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Olive" title="b2">f2</font> holds <br/>( ( <font color="Olive" title="b1">f1</font> is  <a href="waybel_0.html#V17" title="WAYBEL_0:attr.17">infs-preserving</a>  implies <font color="Olive" title="b2">f2</font> is  <a href="waybel_0.html#V17" title="WAYBEL_0:attr.17">infs-preserving</a>  ) &amp; ( <font color="Olive" title="b1">f1</font> is  <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a>  implies <font color="Olive" title="b2">f2</font> is  <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a>  ) ) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:9"/><span class="lab"><font color="Green" title="E6">A1</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="c1">S1</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">S1</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="c3">S2</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c3">S2</font> #)
 <span class="kw">and </span><br/><a NAME="E2:9"/><span class="lab"><font color="Green" title="E7">A2</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="c2">T1</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c2">T1</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">T2</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c4">T2</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">f1</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">T1</font><br/> for <font color="Olive" title="b2">f2</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c3">S2</font>,<font color="Maroon" title="c4">T2</font>  st <font color="Olive" title="b1">f1</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Olive" title="b2">f2</font> holds <br/>( ( <font color="Olive" title="b1">f1</font> is  <a href="waybel_0.html#V17" title="WAYBEL_0:attr.17">infs-preserving</a>  implies <font color="Olive" title="b2">f2</font> is  <a href="waybel_0.html#V17" title="WAYBEL_0:attr.17">infs-preserving</a>  ) &amp; ( <font color="Olive" title="b1">f1</font> is  <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a>  implies <font color="Olive" title="b2">f2</font> is  <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a>  ) )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c5">f1</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">T1</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">f2</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c3">S2</font>,<font color="Maroon" title="c4">T2</font>  st <font color="Maroon" title="c5">f1</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Olive" title="b1">f2</font> holds <br/>( ( <font color="Maroon" title="c5">f1</font> is  <a href="waybel_0.html#V17" title="WAYBEL_0:attr.17">infs-preserving</a>  implies <font color="Olive" title="b1">f2</font> is  <a href="waybel_0.html#V17" title="WAYBEL_0:attr.17">infs-preserving</a>  ) &amp; ( <font color="Maroon" title="c5">f1</font> is  <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a>  implies <font color="Olive" title="b1">f2</font> is  <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a>  ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c6">f2</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c3">S2</font>,<font color="Maroon" title="c4">T2</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="c5">f1</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c6">f2</font> implies ( ( <font color="Maroon" title="c5">f1</font> is  <a href="waybel_0.html#V17" title="WAYBEL_0:attr.17">infs-preserving</a>  implies <font color="Maroon" title="c6">f2</font> is  <a href="waybel_0.html#V17" title="WAYBEL_0:attr.17">infs-preserving</a>  ) &amp; ( <font color="Maroon" title="c5">f1</font> is  <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a>  implies <font color="Maroon" title="c6">f2</font> is  <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a>  ) ) )</span><br/>



<span class="kw">assume </span><a NAME="E3:9"/><span class="lab"><font color="Green" title="E8">A3</font></span>: 
<font color="Maroon" title="c5">f1</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c6">f2</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="c5">f1</font> is  <a href="waybel_0.html#V17" title="WAYBEL_0:attr.17">infs-preserving</a>  implies <font color="Maroon" title="c6">f2</font> is  <a href="waybel_0.html#V17" title="WAYBEL_0:attr.17">infs-preserving</a>  ) &amp; ( <font color="Maroon" title="c5">f1</font> is  <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a>  implies <font color="Maroon" title="c6">f2</font> is  <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a>  ) )</span><br/>

<span class="kw">thus </span><a NAME="E4:9"/>
( <font color="Maroon" title="c5">f1</font> is  <a href="waybel_0.html#V17" title="WAYBEL_0:attr.17">infs-preserving</a>  implies <font color="Maroon" title="c6">f2</font> is  <a href="waybel_0.html#V17" title="WAYBEL_0:attr.17">infs-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="c5">f1</font> is  <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a>  implies <font color="Maroon" title="c6">f2</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="9_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:9_1"/><span class="lab"><font color="Green" title="E9">A4</font></span>: 
for <font color="Olive" title="b1">X</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">S1</font> holds  <font color="Maroon" title="c5">f1</font> <a href="waybel_0.html#R3" title="WAYBEL_0:pred.3">preserves_inf_of</a> <font color="Olive" title="b1">X</font>
 ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="waybel_0.html#D32" title="WAYBEL_0:def.32">WAYBEL_0:def 32</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">f2</font> is  <a href="waybel_0.html#V17" title="WAYBEL_0:attr.17">infs-preserving</a> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c7">X</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">S2</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="waybel_0.html#D32" title="WAYBEL_0:def.32">WAYBEL_0:def 32</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">f2</font> <a href="waybel_0.html#R3" title="WAYBEL_0:pred.3">preserves_inf_of</a> <font color="Maroon" title="c7">X</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c8">Y</font> = <font color="Maroon" title="c7">X</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">S1</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9"><span class="lab"><font color="Green" title="E6">A1</font></span></a></span>;<br/>
<span class="kw">thus </span><a NAME="E3:9_1"/>
<font color="Maroon" title="c6">f2</font> <a href="waybel_0.html#R3" title="WAYBEL_0:pred.3">preserves_inf_of</a> <font color="Maroon" title="c7">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="txt" href="#E2:9"><span class="lab"><font color="Green" title="E7">A2</font></span></a>, <a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="txt" href="#E1:9_1"><span class="lab"><font color="Green" title="E9">A4</font></span></a>, <a class="ref" href="waybel_0.html#T65" title="WAYBEL_0:th.65">WAYBEL_0:65</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">assume </span><a NAME="E5:9"/><span class="lab"><font color="Green" title="E9">A5</font></span>: 
for <font color="Olive" title="b1">X</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">S1</font>  st not <font color="Olive" title="b1">X</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; <font color="Olive" title="b1">X</font> is  <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a>  holds <br/><font color="Maroon" title="c5">f1</font> <a href="waybel_0.html#R4" title="WAYBEL_0:pred.4">preserves_sup_of</a> <font color="Olive" title="b1">X</font>
 ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="waybel_0.html#D37" title="WAYBEL_0:def.37">WAYBEL_0:def 37</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">f2</font> is  <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c7">X</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">S2</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="waybel_0.html#D37" title="WAYBEL_0:def.37">WAYBEL_0:def 37</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="c7">X</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  or  not <font color="Maroon" title="c7">X</font> is  <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a>  or <font color="Maroon" title="c6">f2</font> <a href="waybel_0.html#R4" title="WAYBEL_0:pred.4">preserves_sup_of</a> <font color="Maroon" title="c7">X</font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c8">Y</font> = <font color="Maroon" title="c7">X</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">S1</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9"><span class="lab"><font color="Green" title="E6">A1</font></span></a></span>;<br/>
<span class="kw">assume </span><a NAME="E7:9"/>
( not <font color="Maroon" title="c7">X</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; <font color="Maroon" title="c7">X</font> is  <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</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">f2</font> <a href="waybel_0.html#R4" title="WAYBEL_0:pred.4">preserves_sup_of</a> <font color="Maroon" title="c7">X</font></span><br/>

<a NAME="E8:9"/><span class="kw">then </span>
<font color="Maroon" title="c5">f1</font> <a href="waybel_0.html#R4" title="WAYBEL_0:pred.4">preserves_sup_of</a> <font color="Maroon" title="c8">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="txt" href="#E5:9"><span class="lab"><font color="Green" title="E9">A5</font></span></a>, <a class="ref" href="waybel_0.html#T3" title="WAYBEL_0:th.3">WAYBEL_0:3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:9"/>
<font color="Maroon" title="c6">f2</font> <a href="waybel_0.html#R4" title="WAYBEL_0:pred.4">preserves_sup_of</a> <font color="Maroon" title="c7">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="txt" href="#E2:9"><span class="lab"><font color="Green" title="E7">A2</font></span></a>, <a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="ref" href="waybel_0.html#T65" title="WAYBEL_0:th.65">WAYBEL_0:65</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>
