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<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:45_1_1_1"/>
<font color="Maroon" title="c2">N</font> is <a href="waybel_0.html#V8" title="WAYBEL_0:attr.8">monotone</a>
 ;<br/><a NAME="E2:45_1_1_1"/><b>then </b><i><font color="Green" title="E21">A1</font></i>: 
 <a href="waybel_0.html#K1" title="WAYBEL_0:func.1">netmap</a> <font color="Maroon" title="c2">N</font>,<font color="Maroon" title="c1">R</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a>
 <b>by </b><i><a class="ref" href="waybel_0.html#D9" title="WAYBEL_0:def.9">WAYBEL_0:def 9</a></i>;<br/><b>let </b><font color="Maroon" title="c3">i</font>, <font color="Maroon" title="c4">j</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="c5">i'</font> = <font color="Maroon" title="c3">i</font>, <font color="Maroon" title="c6">j'</font> = <font color="Maroon" title="c4">j</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font> ;<br/><b>assume </b><a NAME="E4:45_1_1_1"/>
<font color="Maroon" title="c3">i</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c4">j</font>
 ;<br/><b>hence </b><a NAME="E5:45_1_1_1"/>
<font color="Maroon" title="c2">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c3">i</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c4">j</font>
 <b>by </b><i><a class="txt" href="waybel11.html#E2:45_1_1_1"><i><font color="Green" title="E21">A1</font></i></a>, <a class="ref" href="waybel_1.html#D2" title="WAYBEL_1:def.2">WAYBEL_1:def 2</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:45_1_1"/><i><font color="Green" title="E21">A2</font></i>: 
for <font color="Olive" title="b1">i</font>, <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 <font color="Olive" title="b1">i</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b2">j</font> holds <br/><font color="Maroon" title="c2">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="c2">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b2">j</font>
 ;<br/>

<a NAME="E3:45_1_1"/>
 <a href="waybel_0.html#K1" title="WAYBEL_0:func.1">netmap</a> <font color="Maroon" title="c2">N</font>,<font color="Maroon" title="c1">R</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_1.html#D2" title="WAYBEL_1:def.2">WAYBEL_1:def 2</a><br/>

<a NAME="E1:45_1_1_2"/><i><font color="Green" title="E22">A3</font></i>: 
<span class="p1">(<span class="default"><a href="waybel_0.html#K1" title="WAYBEL_0:func.1">netmap</a> <font color="Maroon" title="c2">N</font>,<font color="Maroon" title="c1">R</font></span>)</span> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c3">x</font>
 
;<br/>
<a NAME="E2:45_1_1_2"/><i><font color="Green" title="E23">A4</font></i>: 
<span class="p1">(<span class="default"><a href="waybel_0.html#K1" title="WAYBEL_0:func.1">netmap</a> <font color="Maroon" title="c2">N</font>,<font color="Maroon" title="c1">R</font></span>)</span> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c4">y</font>
 
;<br/>
<b>assume </b><a NAME="E3:45_1_1_2"/>
<font color="Maroon" title="c3">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c4">y</font>
 ;<br/>

<b>hence </b><a NAME="E4:45_1_1_2"/>
<span class="p1">(<span class="default"><a href="waybel_0.html#K1" title="WAYBEL_0:func.1">netmap</a> <font color="Maroon" title="c2">N</font>,<font color="Maroon" title="c1">R</font></span>)</span> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Maroon" title="c3">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K1" title="WAYBEL_0:func.1">netmap</a> <font color="Maroon" title="c2">N</font>,<font color="Maroon" title="c1">R</font></span>)</span> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Maroon" title="c4">y</font>
 <b>by </b><i><a class="txt" href="waybel11.html#E2:45_1_1"><i><font color="Green" title="E21">A2</font></i></a>, <a class="txt" href="waybel11.html#E1:45_1_1_2"><i><font color="Green" title="E22">A3</font></i></a>, <a class="txt" href="waybel11.html#E2:45_1_1_2"><i><font color="Green" title="E23">A4</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:45_1_1"/>
<font color="Maroon" title="c2">N</font> is <a href="waybel_0.html#V8" title="WAYBEL_0:attr.8">monotone</a>
 <b>by </b><i><a class="ref" href="waybel_0.html#D9" title="WAYBEL_0:def.9">WAYBEL_0:def 9</a></i>;<br/>


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