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<b>let </b><font color="Maroon" title="c1">L</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a>  <a href="struct_0.html#L1" title="STRUCT_0:struct.1">1-sorted</a> ;<br/><b>let </b><font color="Maroon" title="c2">N</font> be   <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">L</font>;<br/><b>let </b><font color="Maroon" title="c3">p</font> be  <a href="waybel28.html#V1" title="WAYBEL28:attr.1">greater_or_equal_to_id</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">N</font>,<font color="Maroon" title="c2">N</font>;<br/>





<a NAME="E1:20"/>
 ex <font color="Olive" title="b1">f</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="waybel28.html#K1" title="WAYBEL28:func.1">*</a> <font color="Maroon" title="c3">p</font></span>)</span>,<font color="Maroon" title="c2">N</font> st <br/>( the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="waybel28.html#K1" title="WAYBEL28:func.1">*</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c2">N</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b1">f</font> &amp; ( for <font color="Olive" title="b2">m</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font>  ex <font color="Olive" title="b3">n</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="waybel28.html#K1" title="WAYBEL28:func.1">*</a> <font color="Maroon" title="c3">p</font></span>)</span> st <br/>for <font color="Olive" title="b4">k</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="waybel28.html#K1" title="WAYBEL28:func.1">*</a> <font color="Maroon" title="c3">p</font></span>)</span>  st <font color="Olive" title="b3">n</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b4">k</font> holds <br/><font color="Olive" title="b2">m</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b1">f</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Olive" title="b4">k</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><b>proof </b></a><div class="add">

<a NAME="E1:20_1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="waybel28.html#K1" title="WAYBEL28:func.1">*</a> <font color="Maroon" title="c3">p</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="c2">N</font>
 
<b>by </b><i><a class="ref" href="waybel28.html#T7" target="_self" title="WAYBEL28:th.7">Th7</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c4">f</font> = <font color="Maroon" title="c3">p</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="waybel28.html#K1" title="WAYBEL28:func.1">*</a> <font color="Maroon" title="c3">p</font></span>)</span>,<font color="Maroon" title="c2">N</font> ;<br/>
<b>take </b>
<font color="Maroon" title="c4">f</font>
;<br/>

<b>thus </b><a NAME="E3:20_1"/>
the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="waybel28.html#K1" title="WAYBEL28:func.1">*</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c2">N</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font>
 <b>by </b><i><a class="ref" href="waybel28.html#D2" target="_self" title="WAYBEL28:def.2">Def2</a></i>;<br/>

<b>let </b><font color="Maroon" title="c5">m</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font>;<br/>

<b>reconsider </b><font color="Maroon" title="c6">n</font> = <font color="Maroon" title="c5">m</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="waybel28.html#K1" title="WAYBEL28:func.1">*</a> <font color="Maroon" title="c3">p</font></span>)</span> <b>by </b><i><a class="ref" href="waybel28.html#T7" target="_self" title="WAYBEL28:th.7">Th7</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c6">n</font>
;<br/>

<b>let </b><font color="Maroon" title="c7">k</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="waybel28.html#K1" title="WAYBEL28:func.1">*</a> <font color="Maroon" title="c3">p</font></span>)</span>;<br/>

<b>assume </b><a NAME="E5:20_1"/><i><font color="Green" title="E15">A1</font></i>: 
<font color="Maroon" title="c6">n</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c7">k</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c8">k1</font> = <font color="Maroon" title="c7">k</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font> <b>by </b><i><a class="ref" href="waybel28.html#T7" target="_self" title="WAYBEL28:th.7">Th7</a></i>;<br/>
<a NAME="E7:20_1"/>
 <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 <span class="p1">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="waybel28.html#K1" title="WAYBEL28:func.1">*</a> <font color="Maroon" title="c3">p</font></span>)</span>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="waybel28.html#K1" title="WAYBEL28:func.1">*</a> <font color="Maroon" title="c3">p</font></span>)</span> #) <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">N</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c2">N</font> #)
 
<b>by </b><i><a class="ref" href="waybel28.html#D2" target="_self" title="WAYBEL28:def.2">Def2</a></i>;<br/>
<a NAME="E8:20_1"/><b>then </b><i><font color="Green" title="E16">A2</font></i>: 
<font color="Maroon" title="c5">m</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c8">k1</font>
 
<b>by </b><i><a class="txt" href="waybel28.html#E5:20_1"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="yellow_0.html#T1" title="YELLOW_0:th.1">YELLOW_0:1</a></i>;<br/>
<a NAME="E9:20_1"/>
<font color="Maroon" title="c8">k1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c3">p</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Maroon" title="c8">k1</font>
 
<b>by </b><i><a class="ref" href="waybel28.html#D1" target="_self" title="WAYBEL28:def.1">Def1</a></i>;<br/>
<b>hence </b><a NAME="E10:20_1"/>
<font color="Maroon" title="c5">m</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c4">f</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Maroon" title="c7">k</font>
 <b>by </b><i><a class="txt" href="waybel28.html#E8:20_1"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="yellow_0.html#D2" title="YELLOW_0:def.2">YELLOW_0:def 2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:20"/>
<font color="Maroon" title="c2">N</font> <a href="waybel28.html#K1" title="WAYBEL28:func.1">*</a> <font color="Maroon" title="c3">p</font> is    <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Maroon" title="c2">N</font>
 <b>by </b><i><a class="ref" href="yellow_6.html#D12" title="YELLOW_6:def.12">YELLOW_6:def 12</a></i>;<br/>


</div>
