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<b>let </b><font color="Maroon" title="c1">S</font>, <font color="Maroon" title="c2">T</font> be   <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>;<br/><b>let </b><font color="Maroon" title="c3">f</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font>;<br/>



<b>assume </b><a NAME="E1:32"/><i><font color="Green" title="E20">A1</font></i>: 
for <font color="Olive" title="b1">N</font> being  <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">S</font> holds  <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="waybel11.html#K1" title="WAYBEL11:func.1">lim_inf</a> <font color="Olive" title="b1">N</font></span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="waybel11.html#K1" title="WAYBEL11:func.1">lim_inf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="waybel_9.html#K7" title="WAYBEL_9:func.7">*</a> <font color="Olive" title="b1">N</font></span>)</span>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font>;<br/><b>assume </b><a NAME="E1:32_1"/><i><font color="Green" title="E21">A2</font></i>: 
<font color="Maroon" title="c4">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">b</font>
 ;<br/><b>set </b><font color="Maroon" title="c6">N</font> =  <a href="waybel17.html#K3" title="WAYBEL17:func.3">Net-Str</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>;<br/><a NAME="E2:32_1"/><i><font color="Green" title="E22">A3</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="waybel11.html#K1" title="WAYBEL11:func.1">lim_inf</a> <span class="p2">(<span class="default"><a href="waybel17.html#K3" title="WAYBEL17:func.3">Net-Str</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">a</font>
 <b>by </b><i><a class="txt" href="waybel17.html#E1:32_1"><i><font color="Green" title="E21">A2</font></i></a>, <a class="txt" href="waybel17.html#E15"><i><font color="Green" title="E14">Lm5</font></i></a></i>;<br/><a NAME="E3:32_1"/>
 <a href="waybel11.html#K1" title="WAYBEL11:func.1">lim_inf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="waybel_9.html#K7" title="WAYBEL_9:func.7">*</a> <span class="p2">(<span class="default"><a href="waybel17.html#K3" title="WAYBEL17:func.3">Net-Str</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">b</font></span>)</span>
 <b>by </b><i><a class="ref" href="waybel17.html#T8" target="_self" title="WAYBEL17:th.8">Th8</a></i>;<br/><a NAME="E4:32_1"/><b>then </b><i><font color="Green" title="E23">A4</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel17.html#E1:32"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="waybel17.html#E2:32_1"><i><font color="Green" title="E22">A3</font></i></a></i>;<br/><a NAME="E5:32_1"/>
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">a</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">b</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_0.html#T23" title="YELLOW_0:th.23">YELLOW_0:23</a></i>;<br/><a NAME="E6:32_1"/><b>then </b>
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel17.html#E4:32_1"><i><font color="Green" title="E23">A4</font></i></a>, <a class="ref" href="orders_2.html#T25" title="ORDERS_2:th.25">ORDERS_2:25</a></i>;<br/><b>hence </b><a NAME="E7:32_1"/>
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">b</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T25" title="YELLOW_0:th.25">YELLOW_0:25</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:32"/>
<font color="Maroon" title="c3">f</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a>
 <b>by </b><i><a class="ref" href="waybel_1.html#D2" title="WAYBEL_1:def.2">WAYBEL_1:def 2</a></i>;<br/>


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