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<b>let </b><font color="Maroon">S</font> be  <a href="yellow_0.html#V1">lower-bounded</a> <a href="waybel_0.html#V24">up-complete</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/><b>let </b><font color="Maroon">T</font> be  <a href="yellow_0.html#V1">lower-bounded</a> <a href="waybel_3.html#V3">continuous</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>assume </b><a NAME="E1:36"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> holds  <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K1">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">w</font></span>)</span> where <font color="Olive">w</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> : <font color="Olive">w</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Olive">x</font> </span>}</span> ,<font color="Maroon">T</font>
 ;<br/>

<a NAME="E2:36"/><b>then </b><i><font color="Green">E46</font></i>: 
<font color="Maroon">f</font> is <a href="orders_3.html#V5">monotone</a>
 
<b>by </b><i/>;<br/>
<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font>;<br/>



<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:36_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">y</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 ;<br/><b>reconsider </b><font color="Maroon">D</font> = <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">x</font></span>)</span> as  non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V1">directed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> <b>by </b><i>, <a class="ref" href="yellow_2.html#T17">YELLOW_2:17</a></i>;<br/><a NAME="E3:36_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K1">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">w</font></span>)</span> where <font color="Olive">w</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> : <font color="Olive">w</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">x</font> </span>}</span> ,<font color="Maroon">T</font>
 <b>by </b><i/>;<br/><a NAME="E4:36_1"/><i><font color="Green">E49</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>] <b>means </b><font color="Maroon">a<sub>1</sub></font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">x</font>;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>) <b>-&gt; </b>  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> = <font color="Maroon">a<sub>1</sub></font>;<br/><a NAME="E5:36_1"/>
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">w</font>) where <font color="Olive">w</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">w</font>] </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">w</font>)</span>)</span> where <font color="Olive">w</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">w</font>] </span>}</span> 
 <b>from </b><i><a class="ref" href="waybel17.html#S2" target="_self">WAYBEL17:sch 2</a>();<br/></i><a NAME="E6:36_1"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D</font>
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">w</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E8:36_1"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> &amp; <font color="Maroon">y</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">w</font> )
 <b>by </b><i>, <a class="ref" href="waybel_4.html#T54">WAYBEL_4:54</a></i>;<br/><b>consider </b><font color="Maroon">v</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E10:36_1"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">x</font> &amp; <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> )
 <b>by </b><i>, <a class="ref" href="funct_2.html#T115">FUNCT_2:115</a></i>;<br/><b>reconsider </b><font color="Maroon">v</font> = <font color="Maroon">v</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> <b>by </b><i/>;<br/><b>take </b><font color="Maroon">v</font> = <font color="Maroon">v</font>;<br/><b>thus </b><a NAME="E12:36_1"/>
( <font color="Maroon">v</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">y</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> )
 <b>by </b><i>, , <a class="ref" href="waybel_3.html#T7">WAYBEL_3:7</a></i>;<br/></div>
<b>end;</b></div>

<b>given </b><font color="Maroon">w</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font><b> such that </b><a NAME="E4:36"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">w</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">y</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font> )
 ;<br/>

<a NAME="E5:36"/>
<font color="Maroon">w</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x</font>
 
<b>by </b><i>, <a class="ref" href="waybel_3.html#T1">WAYBEL_3:1</a></i>;<br/>
<a NAME="E6:36"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</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="E7:36"/>
<font color="Maroon">y</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="waybel_3.html#T2">WAYBEL_3:2</a></i>;<br/>


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