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

<b>let </b><font color="Maroon" title="c1">L1</font>, <font color="Maroon" title="c2">L2</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="struct_0.html#L1" title="STRUCT_0:struct.1">1-sorted</a> ;<br/>

<b>assume </b><a NAME="E1:12"/><i><font color="Green" title="E8">A1</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L1</font> <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">L2</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c3">N1</font> be   <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">L1</font>;<br/><b>let </b><font color="Maroon" title="c4">N2</font> be   <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c2">L2</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E2:12"/><i><font color="Green" title="E9">A2</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="c3">N1</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c3">N1</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">N2</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c4">N2</font> #)
 <b>and </b><br/><a NAME="E3:12"/><i><font color="Green" title="E10">A3</font></i>: 
the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c3">N1</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c4">N2</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c5">S1</font> be    <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Maroon" title="c3">N1</font>;<br/>

<b>consider </b><font color="Maroon" title="c6">S2</font> being  <a href="waybel_0.html#V6" title="WAYBEL_0:attr.6">strict</a>  <a href="waybel_0.html#L1" title="WAYBEL_0:struct.1">NetStr</a> of <font color="Maroon" title="c2">L2</font><b> such that </b><br/><a NAME="E5:12"/><i><font color="Green" title="E11">A4</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="c5">S1</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c5">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="c6">S2</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c6">S2</font> #)
 <b>and </b><br/><a NAME="E6:12"/><i><font color="Green" title="E12">A5</font></i>: 
the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c5">S1</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c6">S2</font>
 <b>by </b><i><a class="txt" href="waybel33.html#E1:12"><i><font color="Green" title="E8">A1</font></i></a>, <a class="ref" href="waybel33.html#T2" target="_self" title="WAYBEL33:th.2">Th2</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">f</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c5">S1</font>,<font color="Maroon" title="c3">N1</font><b> such that </b><br/><a NAME="E8:12"/><i><font color="Green" title="E13">A6</font></i>: 
the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c5">S1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c3">N1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">f</font>
 <b>and </b><br/><a NAME="E9:12"/><i><font color="Green" title="E14">A7</font></i>: 
for <font color="Olive" title="b1">B5</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">N1</font>  ex <font color="Olive" title="b2">B6</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">S1</font> st <br/>for <font color="Olive" title="b3">B7</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">S1</font>  st <font color="Olive" title="b2">B6</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b3">B7</font> holds <br/><font color="Olive" title="b1">B5</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c7">f</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Olive" title="b3">B7</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/>
<a NAME="E10:12"/>
(  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c5">S1</font> <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="c5">S1</font> &amp;  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c6">S2</font> <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="c6">S2</font> &amp;  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c5">S1</font> is <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> )
 
<b>by </b><i><a class="ref" href="waybel_0.html#D6" title="WAYBEL_0:def.6">WAYBEL_0:def 6</a></i>;<br/>
<a NAME="E11:12"/><b>then </b>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c6">S2</font> is <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a>
 
<b>by </b><i><a class="txt" href="waybel33.html#E5:12"><i><font color="Green" title="E11">A4</font></i></a>, <a class="ref" href="waybel_0.html#T3" title="WAYBEL_0:th.3">WAYBEL_0:3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c8">S2</font> = <font color="Maroon" title="c6">S2</font> as  <a href="waybel_0.html#V6" title="WAYBEL_0:attr.6">strict</a> <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c2">L2</font> <b>by </b><i><a class="txt" href="waybel33.html#E5:12"><i><font color="Green" title="E11">A4</font></i></a>, <a class="ref" href="struct_0.html#D1" title="STRUCT_0:def.1">STRUCT_0:def 1</a>, <a class="ref" href="waybel_0.html#D6" title="WAYBEL_0:def.6">WAYBEL_0:def 6</a>, <a class="ref" href="waybel_8.html#T13" title="WAYBEL_8:th.13">WAYBEL_8:13</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">g</font> = <font color="Maroon" title="c7">f</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c8">S2</font>,<font color="Maroon" title="c4">N2</font> <b>by </b><i><a class="txt" href="waybel33.html#E2:12"><i><font color="Green" title="E9">A2</font></i></a>, <a class="txt" href="waybel33.html#E5:12"><i><font color="Green" title="E11">A4</font></i></a></i>;<br/>
<a NAME="E14:12"/>
<font color="Maroon" title="c8">S2</font> is    <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Maroon" title="c4">N2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon" title="c9">g</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_6.html#D12" title="YELLOW_6:def.12">YELLOW_6:def 12</a><br/>

<b>thus </b><a NAME="E1:12_1"/>
the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c8">S2</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c4">N2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c9">g</font>
 <b>by </b><i><a class="txt" href="waybel33.html#E3:12"><i><font color="Green" title="E10">A3</font></i></a>, <a class="txt" href="waybel33.html#E6:12"><i><font color="Green" title="E12">A5</font></i></a>, <a class="txt" href="waybel33.html#E8:12"><i><font color="Green" title="E13">A6</font></i></a></i>;<br/>

<b>let </b><font color="Maroon" title="c10">B2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">N2</font>;<br/>

<b>reconsider </b><font color="Maroon" title="c11">b2</font> = <font color="Maroon" title="c10">B2</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">N1</font> <b>by </b><i><a class="txt" href="waybel33.html#E2:12"><i><font color="Green" title="E9">A2</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">b6</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">S1</font><b> such that </b><br/><a NAME="E4:12_1"/><i><font color="Green" title="E15">A8</font></i>: 
for <font color="Olive" title="b1">b7</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">S1</font>  st <font color="Maroon" title="c12">b6</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b1">b7</font> holds <br/><font color="Maroon" title="c11">b2</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c7">f</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Olive" title="b1">b7</font>
 <b>by </b><i><a class="txt" href="waybel33.html#E9:12"><i><font color="Green" title="E14">A7</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">B3</font> = <font color="Maroon" title="c12">b6</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c8">S2</font> <b>by </b><i><a class="txt" href="waybel33.html#E5:12"><i><font color="Green" title="E11">A4</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c13">B3</font>
;<br/>

<b>let </b><font color="Maroon" title="c14">B4</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c8">S2</font>;<br/>

<b>reconsider </b><font color="Maroon" title="c15">b4</font> = <font color="Maroon" title="c14">B4</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">S1</font> <b>by </b><i><a class="txt" href="waybel33.html#E5:12"><i><font color="Green" title="E11">A4</font></i></a></i>;<br/>
<b>assume </b><a NAME="E7:12_1"/>
<font color="Maroon" title="c13">B3</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c14">B4</font>
 ;<br/>

<a NAME="E8:12_1"/><b>then </b>
<font color="Maroon" title="c12">b6</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c15">b4</font>
 
<b>by </b><i><a class="txt" href="waybel33.html#E5:12"><i><font color="Green" title="E11">A4</font></i></a>, <a class="ref" href="yellow_0.html#T1" title="YELLOW_0:th.1">YELLOW_0:1</a></i>;<br/>
<a NAME="E9:12_1"/><b>then </b>
<font color="Maroon" title="c11">b2</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c7">f</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Maroon" title="c15">b4</font>
 
<b>by </b><i><a class="txt" href="waybel33.html#E4:12_1"><i><font color="Green" title="E15">A8</font></i></a></i>;<br/>
<b>hence </b><a NAME="E10:12_1"/>
<font color="Maroon" title="c10">B2</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c9">g</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Maroon" title="c14">B4</font>
 <b>by </b><i><a class="txt" href="waybel33.html#E2:12"><i><font color="Green" title="E9">A2</font></i></a>, <a class="ref" href="yellow_0.html#T1" title="YELLOW_0:th.1">YELLOW_0:1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E15:12"/>
 ex <font color="Olive" title="b1">S2</font> being <a href="waybel_0.html#V6" title="WAYBEL_0:attr.6">strict</a>  <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Maroon" title="c4">N2</font> st <br/>(  <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="c5">S1</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c5">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="Olive" title="b1">S2</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Olive" title="b1">S2</font> #) &amp; the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c5">S1</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Olive" title="b1">S2</font> )
 <b>by </b><i><a class="txt" href="waybel33.html#E5:12"><i><font color="Green" title="E11">A4</font></i></a>, <a class="txt" href="waybel33.html#E6:12"><i><font color="Green" title="E12">A5</font></i></a></i>;<br/>


</div>
