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<b>let </b><font color="Maroon">R</font> be   <a href="waybel_0.html#NM8">LATTICE</a>, <font color="Maroon">S</font> be   <a href="waybel_0.html#NM8">LATTICE</a>, <font color="Maroon">T</font> be   <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 <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">R</font>,<font color="Maroon">S</font></span><a href="yellow_3.html#K3">:]</a></span>,<font color="Maroon">T</font>;<br/>



<b>assume </b><a NAME="E1:23"/><i><font color="Green">E27</font></i>: 
for <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font><br/> for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> holds <br/> (  <a href="waybel24.html#K1" target="_self">Proj</a> <font color="Maroon">f</font>,<font color="Olive">a</font> is <a href="orders_3.html#V5">monotone</a> &amp;  <a href="waybel24.html#K2" target="_self">Proj</a> <font color="Maroon">f</font>,<font color="Olive">b</font> is <a href="orders_3.html#V5">monotone</a> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_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"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">R</font>,<font color="Maroon">S</font></span><a href="yellow_3.html#K3">:]</a></span>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">R</font>,<font color="Maroon">S</font></span><a href="yellow_3.html#K3">:]</a></span>;<br/><b>assume </b><a NAME="E1:23_1"/>
<font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font>
 ;<br/><a NAME="E2:23_1"/><b>then </b><i><font color="Green">E43</font></i>: 
( <font color="Maroon">x</font> <a href="yellow_3.html#K8">`1</a>  <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font> <a href="yellow_3.html#K8">`1</a>  &amp; <font color="Maroon">x</font> <a href="yellow_3.html#K9">`2</a>  <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font> <a href="yellow_3.html#K9">`2</a>  )
 <b>by </b><i><a class="ref" href="yellow_3.html#T12">YELLOW_3:12</a></i>;<br/><a NAME="E3:23_1"/><i><font color="Green">E44</font></i>: 
 <a href="waybel24.html#K2" target="_self">Proj</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="yellow_3.html#K9">`2</a> </span>)</span> is <a href="orders_3.html#V5">monotone</a>
 <b>by </b><i/>;<br/><a NAME="E4:23_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="yellow_3.html#K8">`1</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="yellow_3.html#K9">`2</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel24.html#K2" target="_self">Proj</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="yellow_3.html#K9">`2</a> </span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="yellow_3.html#K8">`1</a> </span>)</span>
 <b>by </b><i/>;<br/><a NAME="E5:23_1"/>
<font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="yellow_3.html#K8">`1</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="yellow_3.html#K9">`2</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel24.html#K2" target="_self">Proj</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="yellow_3.html#K9">`2</a> </span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="yellow_3.html#K8">`1</a> </span>)</span>
 <b>by </b><i/>;<br/><a NAME="E6:23_1"/><b>then </b><i><font color="Green">E58</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="yellow_3.html#K8">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="yellow_3.html#K9">`2</a> </span>)</span></span><a href="yellow_3.html#K7">]</a></span> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="yellow_3.html#K8">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="yellow_3.html#K9">`2</a> </span>)</span></span><a href="yellow_3.html#K7">]</a></span>
 <b>by </b><i>, , , <a class="ref" href="waybel_1.html#D2">WAYBEL_1:def 2</a></i>;<br/><a NAME="E7:23_1"/><i><font color="Green">E59</font></i>: 
 <a href="waybel24.html#K1" target="_self">Proj</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="yellow_3.html#K8">`1</a> </span>)</span> is <a href="orders_3.html#V5">monotone</a>
 <b>by </b><i/>;<br/><a NAME="E8:23_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="yellow_3.html#K8">`1</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="yellow_3.html#K9">`2</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel24.html#K1" target="_self">Proj</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="yellow_3.html#K8">`1</a> </span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="yellow_3.html#K9">`2</a> </span>)</span>
 <b>by </b><i/>;<br/><a NAME="E9:23_1"/>
<font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="yellow_3.html#K8">`1</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="yellow_3.html#K9">`2</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel24.html#K1" target="_self">Proj</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="yellow_3.html#K8">`1</a> </span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="yellow_3.html#K9">`2</a> </span>)</span>
 <b>by </b><i/>;<br/><a NAME="E10:23_1"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="yellow_3.html#K8">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="yellow_3.html#K9">`2</a> </span>)</span></span><a href="yellow_3.html#K7">]</a></span> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="yellow_3.html#K8">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="yellow_3.html#K9">`2</a> </span>)</span></span><a href="yellow_3.html#K7">]</a></span>
 <b>by </b><i>, , , <a class="ref" href="waybel_1.html#D2">WAYBEL_1:def 2</a></i>;<br/><a NAME="E11:23_1"/><b>then </b><i><font color="Green">E73</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="yellow_3.html#K8">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="yellow_3.html#K9">`2</a> </span>)</span></span><a href="yellow_3.html#K7">]</a></span> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="yellow_3.html#K8">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="yellow_3.html#K9">`2</a> </span>)</span></span><a href="yellow_3.html#K7">]</a></span>
 <b>by </b><i>, <a class="ref" href="yellow_0.html#D2">YELLOW_0:def 2</a></i>;<br/><a NAME="E12:23_1"/><i><font color="Green">E77</font></i>: 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">R</font>,<font color="Maroon">S</font></span><a href="yellow_3.html#K3">:]</a></span>
 <b>by </b><i><a class="ref" href="yellow_3.html#D2">YELLOW_3:def 2</a></i>;<br/><a NAME="E13:23_1"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="yellow_3.html#K8">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="yellow_3.html#K9">`2</a> </span>)</span></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="mcart_1.html#T23">MCART_1:23</a></i>;<br/><b>hence </b><a NAME="E14:23_1"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>
 <b>by </b><i>, , <a class="ref" href="mcart_1.html#T23">MCART_1:23</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:23"/>
<font color="Maroon">f</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/>


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