<?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="struct_0.html#L1" title="STRUCT_0:struct.1">1-sorted</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">N</font>, <font color="Olive" title="b2">M</font> being  <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">L</font>  st  <a href="waybel_0.html#G1" title="WAYBEL_0:aggr.1">NetStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">N</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Olive" title="b1">N</font>,the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Olive" title="b1">N</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_0.html#G1" title="WAYBEL_0:aggr.1">NetStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b2">M</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Olive" title="b2">M</font>,the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Olive" title="b2">M</font> #) holds <br/><font color="Olive" title="b2">M</font> is    <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Olive" title="b1">N</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">N</font>, <font color="Maroon" title="c3">M</font> be   <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</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"> (  <a href="waybel_0.html#G1" title="WAYBEL_0:aggr.1">NetStr</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>,the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c2">N</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_0.html#G1" title="WAYBEL_0:aggr.1">NetStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">M</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c3">M</font>,the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c3">M</font> #) implies <font color="Maroon" title="c3">M</font> is    <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Maroon" title="c2">N</font> )</span><br/>



<span class="kw">assume </span><a NAME="E1:18"/><span class="lab"><font color="Green" title="E13">A1</font></span>: 
 <a href="waybel_0.html#G1" title="WAYBEL_0:aggr.1">NetStr</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>,the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c2">N</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_0.html#G1" title="WAYBEL_0:aggr.1">NetStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">M</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c3">M</font>,the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c3">M</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="c3">M</font> is    <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Maroon" title="c2">N</font></span><br/>

<a NAME="E2:18"/><span class="lab"><font color="Green" title="E14">A2</font></span>: 
<font color="Maroon" title="c2">N</font> is    <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Maroon" title="c2">N</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_6.html#T23" title="YELLOW_6:th.23">YELLOW_6:23</a></span>;<br/>
<a NAME="E3:18"/>
 ex <font color="Olive" title="b1">f</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c2">N</font> st <br/>( the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c3">M</font> <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 <font color="Maroon" title="c3">M</font> st <br/>for <font color="Olive" title="b4">p</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">M</font>  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">p</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#K2" title="YELLOW_6:func.2">.</a> <font color="Olive" title="b4">p</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">consider </span><font color="Maroon" title="c4">f</font> being   <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><span class="kw"> such that </span><br/><a NAME="E2:18_1"/><span class="lab"><font color="Green" title="E15">A3</font></span>: 
the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c2">N</font> <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>
 <span class="kw">and </span><br/><a NAME="E3:18_1"/><span class="lab"><font color="Green" title="E16">A4</font></span>: 
for <font color="Olive" title="b1">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="b2">n</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font> st <br/>for <font color="Olive" title="b3">p</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font>  st <font color="Olive" title="b2">n</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b3">p</font> holds <br/><font color="Olive" title="b1">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#K2" title="YELLOW_6:func.2">.</a> <font color="Olive" title="b3">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:18"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="ref" href="yellow_6.html#D12" title="YELLOW_6:def.12">YELLOW_6:def 12</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">f2</font> = <font color="Maroon" title="c4">f</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c2">N</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18"><span class="lab"><font color="Green" title="E13">A1</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c5">f2</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c3">M</font> <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="c5">f2</font> &amp; ( for <font color="Olive" title="b1">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="b2">n</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">M</font> st <br/>for <font color="Olive" title="b3">p</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">M</font>  st <font color="Olive" title="b2">n</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b3">p</font> holds <br/><font color="Olive" title="b1">m</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">f2</font> <a href="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Olive" title="b3">p</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E5:18_1"/>
the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c3">M</font> <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="c5">f2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18"><span class="lab"><font color="Green" title="E13">A1</font></span></a>, <a class="txt" href="#E2:18_1"><span class="lab"><font color="Green" title="E15">A3</font></span></a></span>; <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">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="b2">n</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">M</font> st <br/>for <font color="Olive" title="b3">p</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">M</font>  st <font color="Olive" title="b2">n</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b3">p</font> holds <br/><font color="Olive" title="b1">m</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">f2</font> <a href="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Olive" title="b3">p</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c6">m</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">n</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">M</font> st <br/>for <font color="Olive" title="b2">p</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">M</font>  st <font color="Olive" title="b1">n</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b2">p</font> holds <br/><font color="Maroon" title="c6">m</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">f2</font> <a href="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Olive" title="b2">p</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c7">n</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font><span class="kw"> such that </span><br/><a NAME="E7:18_1"/><span class="lab"><font color="Green" title="E17">A5</font></span>: 
for <font color="Olive" title="b1">p</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font>  st <font color="Maroon" title="c7">n</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> holds <br/><font color="Maroon" title="c6">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#K2" title="YELLOW_6:func.2">.</a> <font color="Olive" title="b1">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:18_1"><span class="lab"><font color="Green" title="E16">A4</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">n2</font> = <font color="Maroon" title="c7">n</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">M</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18"><span class="lab"><font color="Green" title="E13">A1</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c8">n2</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">p</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">M</font>  st <font color="Maroon" title="c8">n2</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> holds <br/><font color="Maroon" title="c6">m</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">f2</font> <a href="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Olive" title="b1">p</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c9">p</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">M</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="c8">n2</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c9">p</font> implies <font color="Maroon" title="c6">m</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">f2</font> <a href="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Maroon" title="c9">p</font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c10">p1</font> = <font color="Maroon" title="c9">p</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18"><span class="lab"><font color="Green" title="E13">A1</font></span></a></span>;<br/>
<span class="kw">assume </span><a NAME="E10:18_1"/>
<font color="Maroon" title="c8">n2</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c9">p</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="c6">m</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">f2</font> <a href="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Maroon" title="c9">p</font></span><br/>

<a NAME="E11:18_1"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">n2</font>,<font color="Maroon" title="c9">p</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c3">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></span>;<br/>
<a NAME="E12:18_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">n</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c10">p1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18"><span class="lab"><font color="Green" title="E13">A1</font></span></a>, <a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:18_1"/>
<font color="Maroon" title="c6">m</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">f2</font> <a href="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Maroon" title="c9">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:18_1"><span class="lab"><font color="Green" title="E17">A5</font></span></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="E4:18"/>
<font color="Maroon" title="c3">M</font> is    <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Maroon" title="c2">N</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_6.html#D12" title="YELLOW_6:def.12">YELLOW_6:def 12</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>
