<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be  <a href="waybel_0.html#V24">up-complete</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="waybel_0.html#V8">monotone</a>  <a href="waybel_0.html#L1">NetStr</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>





<b>assume </b><a NAME="E1:41"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="orders_3.html#V5">monotone</a>
 ;<br/>

<a NAME="E2:41"/><i><font color="Green">E28</font></i>: 
 <a href="waybel_0.html#K1">netmap</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font> is <a href="orders_3.html#V5">monotone</a>
 
<b>by </b><i><a class="ref" href="waybel_0.html#D9">WAYBEL_0:def 9</a></i>;<br/>
<i><font color="Green">E29</font></i>:  <a href="waybel_0.html#K1">netmap</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font> = 
the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>

<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K1">*</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="waybel_9.html#D8">WAYBEL_9:def 8</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K1">netmap</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>

;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="waybel_0.html#K1">netmap</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>;<br/><b>assume </b><a NAME="E1:41_2"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/><a NAME="E2:41_2"/><i><font color="Green">E31</font></i>: 
 <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>4</sub></font> #) <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> #)
 <b>by </b><i><a class="ref" href="waybel_9.html#D8">WAYBEL_9:def 8</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>4</sub></font> ;<br/><a NAME="E4:41_2"/><i><font color="Green">E32</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K1">netmap</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel17.html#E2:41_2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E5:41_2"/><b>then </b><i><font color="Green">E33</font></i>: 
<span class="p1">(<span class="default"><a href="waybel_0.html#K1">netmap</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K1">netmap</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel17.html#E3:41"><i><font color="Green">E29</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/><a NAME="E6:41_2"/><i><font color="Green">E34</font></i>: 
<span class="p1">(<span class="default"><a href="waybel_0.html#K1">netmap</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K1">netmap</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel17.html#E3:41"><i><font color="Green">E29</font></i></a>, <a class="txt" href="waybel17.html#E4:41_2"><i><font color="Green">E32</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/><a NAME="E7:41_2"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel17.html#E1:41_2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/><a NAME="E8:41_2"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="waybel17.html#E2:41_2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/><a NAME="E9:41_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="waybel_0.html#K1">netmap</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K1">netmap</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="waybel17.html#E2:41"><i><font color="Green">E28</font></i></a>, <a class="ref" href="waybel_1.html#D2">WAYBEL_1:def 2</a></i>;<br/><b>hence </b><a NAME="E10:41_2"/>
<span class="p1">(<span class="default"><a href="waybel_0.html#K1">netmap</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K1">netmap</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="waybel17.html#E1:41"><i><font color="Green">E27</font></i></a>, <a class="txt" href="waybel17.html#E5:41_2"><i><font color="Green">E33</font></i></a>, <a class="txt" href="waybel17.html#E6:41_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="waybel_1.html#D2">WAYBEL_1:def 2</a></i>;<br/></div><b>end;</b></div>
<a NAME="E5:41"/><b>then </b>
 <a href="waybel_0.html#K1">netmap</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font> is <a href="orders_3.html#V5">monotone</a>
 
<b>by </b><i><a class="ref" href="waybel_1.html#D2">WAYBEL_1:def 2</a></i>;<br/>
<b>hence </b><a NAME="E6:41"/>
<font color="Maroon">c<sub>3</sub></font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">c<sub>4</sub></font> is <a href="waybel_0.html#V8">monotone</a>
 <b>by </b><i><a class="ref" href="waybel_0.html#D9">WAYBEL_0:def 9</a></i>;<br/>


</div>
