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<b>let </b><font color="Maroon">S</font> be  <a href="waybel_0.html#V24">up-complete</a> <a href="waybel_0.html#NM8">LATTICE</a>, <font color="Maroon">T</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">f</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">S</font>,<font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">N</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">S</font>;<br/>





<b>assume </b><a NAME="E1:41"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">f</font> is <a href="orders_3.html#V5">monotone</a>
 ;<br/>

<a NAME="E2:41"/><i><font color="Green">E46</font></i>: 
 <a href="waybel_0.html#K1">netmap</a> <font color="Maroon">N</font>,<font color="Maroon">S</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/>
<a NAME="E3:41"/><i><font color="Green">E47</font></i>: 
 <a href="waybel_0.html#K1">netmap</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">N</font></span>)</span>,<font color="Maroon">T</font> <a href="funct_2.html#R1">=</a> <font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K1">netmap</a> <font color="Maroon">N</font>,<font color="Maroon">S</font></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel_9.html#D8">WAYBEL_9:def 8</a></i>;<br/>
<b>set </b><font color="Maroon">g</font> =  <a href="waybel_0.html#K1">netmap</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">N</font></span>)</span>,<font color="Maroon">T</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">N</font></span>)</span>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">N</font></span>)</span>;<br/><b>assume </b><a NAME="E1:41_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">y</font>
 ;<br/><a NAME="E2:41_1"/><i><font color="Green">E49</font></i>: 
 <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">N</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">f</font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">N</font></span>)</span>,the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">N</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">x'</font> = <font color="Maroon">x</font>, <font color="Maroon">y'</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> ;<br/><a NAME="E4:41_1"/><i><font color="Green">E52</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">N</font>,<font color="Maroon">S</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">f</font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">N</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E5:41_1"/><b>then </b><i><font color="Green">E53</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">f</font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">N</font></span>)</span>,<font color="Maroon">T</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</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">N</font>,<font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/><a NAME="E6:41_1"/><i><font color="Green">E54</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">f</font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">N</font></span>)</span>,<font color="Maroon">T</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</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">N</font>,<font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/><a NAME="E7:41_1"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</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">f</font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">N</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/><a NAME="E8:41_1"/><b>then </b>
<font color="Maroon">x'</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">y'</font>
 <b>by </b><i>, <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/><a NAME="E9:41_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="waybel_0.html#K1">netmap</a> <font color="Maroon">N</font>,<font color="Maroon">S</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x'</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">N</font>,<font color="Maroon">S</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y'</font>
 <b>by </b><i>, <a class="ref" href="waybel_1.html#D2">WAYBEL_1:def 2</a></i>;<br/><b>hence </b><a NAME="E10:41_1"/>
<span class="p1">(<span class="default"><a href="waybel_0.html#K1">netmap</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">N</font></span>)</span>,<font color="Maroon">T</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</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">f</font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">N</font></span>)</span>,<font color="Maroon">T</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>
 <b>by </b><i>, , , <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">f</font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">N</font></span>)</span>,<font color="Maroon">T</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">f</font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">N</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/>


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