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<b>let </b><font color="Maroon">R</font> be  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#V2">reflexive</a>  <a href="orders_2.html#L1">RelStr</a> , <font color="Maroon">S</font> be  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#V2">reflexive</a>  <a href="orders_2.html#L1">RelStr</a> , <font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#V2">reflexive</a>  <a href="orders_2.html#L1">RelStr</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>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>;<br/>







<b>reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> ;<br/>
<b>set </b><font color="Maroon">g</font> =  <a href="waybel24.html#K2" target="_self">Proj</a> <font color="Maroon">f</font>,<font color="Maroon">b</font>;<br/>
<b>set </b><font color="Maroon">h</font> =  <a href="waybel24.html#K1" target="_self">Proj</a> <font color="Maroon">f</font>,<font color="Maroon">a</font>;<br/>
<b>assume </b><a NAME="E2:18"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">f</font> is <a href="waybel_0.html#V5">antitone</a>
 ;<br/>

<div><i><font color="Green">E43</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_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 <font color="Maroon">S</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>;<br/><b>assume </b><a NAME="E1:18_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font>
 ;<br/><a NAME="E2:18_1"/><i><font color="Green">E53</font></i>: 
<span class="p1">(<span class="default"><a href="waybel24.html#K1" target="_self">Proj</a> <font color="Maroon">f</font>,<font color="Maroon">a</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="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">x</font>
 <b>by </b><i/>;<br/><a NAME="E3:18_1"/><i><font color="Green">E58</font></i>: 
<span class="p1">(<span class="default"><a href="waybel24.html#K1" target="_self">Proj</a> <font color="Maroon">f</font>,<font color="Maroon">a</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="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">y</font>
 <b>by </b><i/>;<br/><a NAME="E4:18_1"/>
<font color="Maroon">a'</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">a'</font>
 ;<br/><a NAME="E5:18_1"/><b>then </b>
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">a'</font>,<font color="Maroon">x</font></span><a href="yellow_3.html#K7">]</a></span> <a href="orders_2.html#R3">&lt;=</a> <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">a'</font>,<font color="Maroon">y</font></span><a href="yellow_3.html#K7">]</a></span>
 <b>by </b><i>, <a class="ref" href="yellow_3.html#T11">YELLOW_3:11</a></i>;<br/><b>hence </b><a NAME="E6:18_1"/>
<span class="p1">(<span class="default"><a href="waybel24.html#K1" target="_self">Proj</a> <font color="Maroon">f</font>,<font color="Maroon">a</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="waybel24.html#K1" target="_self">Proj</a> <font color="Maroon">f</font>,<font color="Maroon">a</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>
 <b>by </b><i>, , , <a class="ref" href="waybel_0.html#D5">WAYBEL_0:def 5</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2"><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 <font color="Maroon">R</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>;<br/><b>assume </b><a NAME="E1:18_2"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font>
 ;<br/><a NAME="E2:18_2"/><i><font color="Green">E60</font></i>: 
<span class="p1">(<span class="default"><a href="waybel24.html#K2" target="_self">Proj</a> <font color="Maroon">f</font>,<font color="Maroon">b</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="binop_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">b</font>
 <b>by </b><i/>;<br/><a NAME="E3:18_2"/><i><font color="Green">E73</font></i>: 
<span class="p1">(<span class="default"><a href="waybel24.html#K2" target="_self">Proj</a> <font color="Maroon">f</font>,<font color="Maroon">b</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="binop_1.html#K1">.</a> <font color="Maroon">y</font>,<font color="Maroon">b</font>
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">b'</font> = <font color="Maroon">b</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> ;<br/><a NAME="E5:18_2"/>
<font color="Maroon">b'</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">b'</font>
 ;<br/><a NAME="E6:18_2"/><b>then </b>
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">b'</font></span><a href="yellow_3.html#K7">]</a></span> <a href="orders_2.html#R3">&lt;=</a> <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">b'</font></span><a href="yellow_3.html#K7">]</a></span>
 <b>by </b><i>, <a class="ref" href="yellow_3.html#T11">YELLOW_3:11</a></i>;<br/><b>hence </b><a NAME="E7:18_2"/>
<span class="p1">(<span class="default"><a href="waybel24.html#K2" target="_self">Proj</a> <font color="Maroon">f</font>,<font color="Maroon">b</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="waybel24.html#K2" target="_self">Proj</a> <font color="Maroon">f</font>,<font color="Maroon">b</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>
 <b>by </b><i>, , , <a class="ref" href="waybel_0.html#D5">WAYBEL_0:def 5</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:18"/>
(  <a href="waybel24.html#K1" target="_self">Proj</a> <font color="Maroon">f</font>,<font color="Maroon">a</font> is <a href="waybel_0.html#V5">antitone</a> &amp;  <a href="waybel24.html#K2" target="_self">Proj</a> <font color="Maroon">f</font>,<font color="Maroon">b</font> is <a href="waybel_0.html#V5">antitone</a> )
 <b>by </b><i>, <a class="ref" href="waybel_9.html#D1">WAYBEL_9:def 1</a></i>;<br/>


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