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<b>let </b><font color="Maroon" title="c1">T</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>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">T</font>;<br/><b>let </b><font color="Maroon" title="c3">M</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="yellow_6.html#V2" title="YELLOW_6:attr.2">full</a>  <a href="yellow_6.html#M1" title="YELLOW_6:mode.1">SubNetStr</a> of <font color="Maroon" title="c2">N</font>;<br/>





<b>assume </b><a NAME="E1:49"/><i><font color="Green" title="E29">A1</font></i>: 
for <font color="Olive" title="b1">i</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">j</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/>( <font color="Olive" title="b2">j</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b2">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">M</font> )
 ;<br/>

<a NAME="E2:49"/><i><font color="Green" title="E30">A2</font></i>: 
( the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c3">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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#K2" title="PARTFUN1:func.2">|</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c3">M</font> is  <a href="yellow_0.html#V4" title="YELLOW_0:attr.4">full</a>  <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of <font color="Maroon" title="c2">N</font> )
 
<b>by </b><i><a class="ref" href="yellow_6.html#D8" title="YELLOW_6:def.8">YELLOW_6:def 8</a>, <a class="ref" href="yellow_6.html#D9" title="YELLOW_6:def.9">YELLOW_6:def 9</a></i>;<br/>
<a NAME="E3:49"/><b>then </b><i><font color="Green" title="E31">A3</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="yellow_0.html#D13" title="YELLOW_0:def.13">YELLOW_0:def 13</a></i>;<br/>
<a NAME="E4:49"/>
<font color="Maroon" title="c3">M</font> is <a href="waybel_0.html#V7" title="WAYBEL_0:attr.7">directed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">M</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_6.html#D5" title="YELLOW_6:def.5">YELLOW_6:def 5</a><br/>

<b>reconsider </b><font color="Maroon" title="c6">i</font> = <font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c7">j</font> = <font color="Maroon" title="c5">y</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="txt" href="waybel21.html#E2:49"><i><font color="Green" title="E30">A2</font></i></a>, <a class="ref" href="yellow_0.html#T59" title="YELLOW_0:th.59">YELLOW_0:59</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">k</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font><b> such that </b><br/><a NAME="E3:49_1"/><i><font color="Green" title="E32">A4</font></i>: 
( <font color="Maroon" title="c8">k</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c6">i</font> &amp; <font color="Maroon" title="c8">k</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c7">j</font> )
 <b>by </b><i><a class="ref" href="yellow_6.html#D5" title="YELLOW_6:def.5">YELLOW_6:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">l</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font><b> such that </b><br/><a NAME="E5:49_1"/><i><font color="Green" title="E33">A5</font></i>: 
( <font color="Maroon" title="c9">l</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c8">k</font> &amp; <font color="Maroon" title="c9">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">M</font> )
 <b>by </b><i><a class="txt" href="waybel21.html#E1:49"><i><font color="Green" title="E29">A1</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">z</font> = <font color="Maroon" title="c9">l</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">M</font> <b>by </b><i><a class="txt" href="waybel21.html#E5:49_1"><i><font color="Green" title="E33">A5</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c10">z</font>
;<br/>

<a NAME="E7:49_1"/>
( <font color="Maroon" title="c9">l</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c6">i</font> &amp; <font color="Maroon" title="c9">l</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c7">j</font> )
 
<b>by </b><i><a class="txt" href="waybel21.html#E3:49_1"><i><font color="Green" title="E32">A4</font></i></a>, <a class="txt" href="waybel21.html#E5:49_1"><i><font color="Green" title="E33">A5</font></i></a>, <a class="ref" href="yellow_0.html#D2" title="YELLOW_0:def.2">YELLOW_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E8:49_1"/>
( <font color="Maroon" title="c4">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c10">z</font> &amp; <font color="Maroon" title="c5">y</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c10">z</font> )
 <b>by </b><i><a class="ref" href="yellow_6.html#T21" title="YELLOW_6:th.21">YELLOW_6:21</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c4">K</font> = <font color="Maroon" title="c3">M</font> as   <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">T</font> <b>by </b><i><a class="txt" href="waybel21.html#E2:49"><i><font color="Green" title="E30">A2</font></i></a></i>;<br/>
<div><i><font color="Green" title="E32">A6</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="49_2"><b>now </b></a><div class="add"><b>set </b><font color="Maroon" title="c5">f</font> =  <a href="yellow_9.html#K1" title="YELLOW_9:func.1">incl</a> <font color="Maroon" title="c4">K</font>,<font color="Maroon" title="c2">N</font>;<br/><a NAME="E1:49_2"/><i><font color="Green" title="E33">A7</font></i>: 
 <a href="yellow_9.html#K1" title="YELLOW_9:func.1">incl</a> <font color="Maroon" title="c4">K</font>,<font color="Maroon" title="c2">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">K</font>
 <b>by </b><i><a class="txt" href="waybel21.html#E3:49"><i><font color="Green" title="E31">A3</font></i></a>, <a class="ref" href="yellow_9.html#D1" title="YELLOW_9:def.1">YELLOW_9:def 1</a></i>;<br/><b>hence </b><a NAME="E2:49_2"/>
the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c4">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <span class="p1">(<span class="default"><a href="yellow_9.html#K1" title="YELLOW_9:func.1">incl</a> <font color="Maroon" title="c4">K</font>,<font color="Maroon" title="c2">N</font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel21.html#E2:49"><i><font color="Green" title="E30">A2</font></i></a>, <a class="ref" href="relat_1.html#T94" title="RELAT_1:th.94">RELAT_1:94</a></i>;<br/><b>let </b><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>;<br/><b>consider </b><font color="Maroon" title="c7">j</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font><b> such that </b><br/><a NAME="E4:49_2"/><i><font color="Green" title="E34">A8</font></i>: 
( <font color="Maroon" title="c7">j</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c6">m</font> &amp; <font color="Maroon" title="c7">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">K</font> )
 <b>by </b><i><a class="txt" href="waybel21.html#E1:49"><i><font color="Green" title="E29">A1</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c8">n</font> = <font color="Maroon" title="c7">j</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">K</font> <b>by </b><i><a class="txt" href="waybel21.html#E4:49_2"><i><font color="Green" title="E34">A8</font></i></a></i>;<br/><b>take </b><font color="Maroon" title="c9">n</font> = <font color="Maroon" title="c8">n</font>;<br/><b>let </b><font color="Maroon" title="c10">p</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">K</font>;<br/><b>reconsider </b><font color="Maroon" title="c11">q</font> = <font color="Maroon" title="c10">p</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="txt" href="waybel21.html#E2:49"><i><font color="Green" title="E30">A2</font></i></a>, <a class="ref" href="yellow_0.html#T59" title="YELLOW_0:th.59">YELLOW_0:59</a></i>;<br/><b>assume </b><a NAME="E7:49_2"/>
<font color="Maroon" title="c9">n</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c10">p</font>
 ;<br/><a NAME="E8:49_2"/><b>then </b>
( <font color="Maroon" title="c7">j</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c11">q</font> &amp; <span class="p1">(<span class="default"><a href="yellow_9.html#K1" title="YELLOW_9:func.1">incl</a> <font color="Maroon" title="c4">K</font>,<font color="Maroon" title="c2">N</font></span>)</span> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Maroon" title="c10">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">q</font> )
 <b>by </b><i><a class="txt" href="waybel21.html#E1:49_2"><i><font color="Green" title="E33">A7</font></i></a>, <a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a>, <a class="ref" href="yellow_6.html#T20" title="YELLOW_6:th.20">YELLOW_6:20</a></i>;<br/><b>hence </b><a NAME="E9:49_2"/>
<font color="Maroon" title="c6">m</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="yellow_9.html#K1" title="YELLOW_9:func.1">incl</a> <font color="Maroon" title="c4">K</font>,<font color="Maroon" title="c2">N</font></span>)</span> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Maroon" title="c10">p</font>
 <b>by </b><i><a class="txt" href="waybel21.html#E4:49_2"><i><font color="Green" title="E34">A8</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="E7:49"/>
<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>
 <b>by </b><i><a class="ref" href="yellow_6.html#D12" title="YELLOW_6:def.12">YELLOW_6:def 12</a></i>;<br/>

<b>hence </b><a NAME="E8:49"/>
 <a href="yellow_9.html#K1" title="YELLOW_9:func.1">incl</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c2">N</font> is    <a href="waybel21.html#M2" title="WAYBEL21:mode.2">Embedding</a> of <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c2">N</font>
 <b>by </b><i><a class="txt" href="waybel21.html#E6:49"><i><font color="Green" title="E32">A6</font></i></a>, <a class="ref" href="waybel21.html#D3" target="_self" title="WAYBEL21:def.3">Def3</a></i>;<br/>


</div>
