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<b>let </b><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#V3" title="STRUCT_0:attr.3">empty</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ;<br/>

<b>assume </b><a NAME="E1:10"/><i><font color="Green" title="E6">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">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> #) )
 ;<br/>

<b>let </b><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>;<br/><b>let </b><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>;<br/>



<b>assume </b><a NAME="E2:10"/><i><font color="Green" title="E7">A2</font></i>: 
<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>
 ;<br/>

<b>thus </b><a NAME="E3:10"/>
( <font color="Maroon" title="c5">f1</font> is <a href="waybel_0.html#V18" title="WAYBEL_0:attr.18">sups-preserving</a> implies <font color="Maroon" title="c6">f2</font> is <a href="waybel_0.html#V18" title="WAYBEL_0:attr.18">sups-preserving</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:10_1"/><i><font color="Green" title="E8">A3</font></i>: 
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#R4" title="WAYBEL_0:pred.4">preserves_sup_of</a> <font color="Olive" title="b1">X</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D33" title="WAYBEL_0:def.33">WAYBEL_0:def 33</a><br/>

<b>let </b><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>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D33" title="WAYBEL_0:def.33">WAYBEL_0:def 33</a><br/>

<b>reconsider </b><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> <b>by </b><i><a class="txt" href="waybel21.html#E1:10"><i><font color="Green" title="E6">A1</font></i></a></i>;<br/>
<a NAME="E3:10_1"/>
<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>
 
<b>by </b><i><a class="txt" href="waybel21.html#E1:10_1"><i><font color="Green" title="E8">A3</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:10_1"/>
<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>
 <b>by </b><i><a class="txt" href="waybel21.html#E1:10"><i><font color="Green" title="E6">A1</font></i></a>, <a class="txt" href="waybel21.html#E2:10"><i><font color="Green" title="E7">A2</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>assume </b><a NAME="E4:10"/><i><font color="Green" title="E8">A4</font></i>: 
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#V2" title="WAYBEL_0:attr.2">filtered</a> holds <br/><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>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D36" title="WAYBEL_0:def.36">WAYBEL_0:def 36</a><br/>

<b>let </b><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>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D36" title="WAYBEL_0:def.36">WAYBEL_0:def 36</a><br/>

<b>reconsider </b><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> <b>by </b><i><a class="txt" href="waybel21.html#E1:10"><i><font color="Green" title="E6">A1</font></i></a></i>;<br/>
<b>assume </b><a NAME="E6:10"/>
( 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#V2" title="WAYBEL_0:attr.2">filtered</a> )
 ;<br/>

<a NAME="E7:10"/><b>then </b>
( not <font color="Maroon" title="c8">Y</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; <font color="Maroon" title="c8">Y</font> is <a href="waybel_0.html#V2" title="WAYBEL_0:attr.2">filtered</a> )
 
<b>by </b><i><a class="txt" href="waybel21.html#E1:10"><i><font color="Green" title="E6">A1</font></i></a>, <a class="ref" href="waybel_0.html#T4" title="WAYBEL_0:th.4">WAYBEL_0:4</a></i>;<br/>
<a NAME="E8:10"/><b>then </b>
<font color="Maroon" title="c5">f1</font> <a href="waybel_0.html#R3" title="WAYBEL_0:pred.3">preserves_inf_of</a> <font color="Maroon" title="c8">Y</font>
 
<b>by </b><i><a class="txt" href="waybel21.html#E4:10"><i><font color="Green" title="E8">A4</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:10"/>
<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>
 <b>by </b><i><a class="txt" href="waybel21.html#E1:10"><i><font color="Green" title="E6">A1</font></i></a>, <a class="txt" href="waybel21.html#E2:10"><i><font color="Green" title="E7">A2</font></i></a>, <a class="ref" href="waybel_0.html#T65" title="WAYBEL_0:th.65">WAYBEL_0:65</a></i>;<br/>


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