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<b>let </b><font color="Maroon" title="c1">L</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="orders_2.html#V3" title="ORDERS_2:attr.3">reflexive</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ;<br/><b>let </b><font color="Maroon" title="c2">D</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font>;<br/><b>let </b><font color="Maroon" title="c3">n</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">D</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>;<br/><b>let </b><font color="Maroon" title="c4">N</font> be   <a href="waybel_0.html#NM1" title="WAYBEL_0:NM.1">prenet</a> of <font color="Maroon" title="c1">L</font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:27"/><i><font color="Green" title="E18">A1</font></i>: 
<font color="Maroon" title="c3">n</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon" title="c2">D</font>
 <b>and </b><br/><a NAME="E2:27"/><i><font color="Green" title="E19">A2</font></i>: 
<font color="Maroon" title="c4">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>(# <font color="Maroon" title="c2">D</font>,<span class="p1">(<span class="default">the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">L</font> <a href="toler_1.html#K1" title="TOLER_1:func.1">|_2</a> <font color="Maroon" title="c2">D</font></span>)</span>,<font color="Maroon" title="c3">n</font> #)
 ;<br/>

<a NAME="E3:27"/>
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="c4">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="c4">N</font> st <br/>for <font color="Olive" title="b3">k</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">N</font>  st <font color="Olive" title="b2">j</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b3">k</font> holds <br/><font color="Maroon" title="c4">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b1">i</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b3">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">i</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">N</font>;<br/>

<b>take </b><font color="Maroon" title="c6">j</font> = <font color="Maroon" title="c5">i</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 <font color="Maroon" title="c4">N</font>;<br/>

<b>assume </b><a NAME="E1:27_1"/><i><font color="Green" title="E20">A3</font></i>: 
<font color="Maroon" title="c6">j</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">nj</font> = <font color="Maroon" title="c3">n</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font>, <font color="Maroon" title="c9">nk</font> = <font color="Maroon" title="c3">n</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <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="c1">L</font> <b>by </b><i><a class="txt" href="waybel_2.html#E2:27"><i><font color="Green" title="E19">A2</font></i></a>, <a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></i>;<br/>
<a NAME="E3:27_1"/><i><font color="Green" title="E21">A4</font></i>: 
<font color="Maroon" title="c4">N</font> is    <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of <font color="Maroon" title="c1">L</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:27_1_1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">N</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="c1">L</font>
 <b>by </b><i><a class="txt" href="waybel_2.html#E2:27"><i><font color="Green" title="E19">A2</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_0.html#D13" title="YELLOW_0:def.13">YELLOW_0:def 13</a><br/>

<b>thus </b><a NAME="E2:27_1_1"/>
the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c4">N</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="txt" href="waybel_2.html#E2:27"><i><font color="Green" title="E19">A2</font></i></a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:27_1"/>
( <font color="Maroon" title="c8">nj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">j</font> &amp; <font color="Maroon" title="c9">nk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">k</font> )
 
<b>by </b><i><a class="txt" href="waybel_2.html#E1:27"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="waybel_2.html#E2:27"><i><font color="Green" title="E19">A2</font></i></a>, <a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></i>;<br/>
<b>hence </b><a NAME="E5:27_1"/>
<font color="Maroon" title="c4">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c5">i</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c7">k</font>
 <b>by </b><i><a class="txt" href="waybel_2.html#E2:27"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="waybel_2.html#E1:27_1"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="waybel_2.html#E3:27_1"><i><font color="Green" title="E21">A4</font></i></a>, <a class="ref" href="yellow_0.html#T60" title="YELLOW_0:th.60">YELLOW_0:60</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:27"/>
<font color="Maroon" title="c4">N</font> is <a href="waybel_0.html#V10" title="WAYBEL_0:attr.10">eventually-directed</a>
 <b>by </b><i><a class="ref" href="waybel_0.html#T11" title="WAYBEL_0:th.11">WAYBEL_0:11</a></i>;<br/>


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