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<b>let </b><font color="Maroon">S</font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel11.html#V4">Scott</a> <a href="waybel_9.html#NM1">TopLattice</a>, <font color="Maroon">T</font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel11.html#V4">Scott</a> <a href="waybel_9.html#NM1">TopLattice</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="E1:59"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">S</font> is <a href="waybel_8.html#V2">algebraic</a> &amp; <font color="Maroon">T</font> is <a href="waybel_8.html#V2">algebraic</a> )
 ;<br/>

<a NAME="E2:59"/><b>then </b><i><font color="Green">E46</font></i>: 
( <font color="Maroon">S</font> is <a href="waybel_3.html#V3">continuous</a> &amp; <font color="Maroon">T</font> is <a href="waybel_3.html#V3">continuous</a> )
 
<b>by </b><i><a class="ref" href="waybel_8.html#T7">WAYBEL_8:7</a></i>;<br/>
<b>assume </b><a NAME="E3:59"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font><br/> for <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> holds <br/> ( <font color="Olive">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="Olive">x</font> iff  ex <font color="Olive">w</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> st <br/>( <font color="Olive">w</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Olive">x</font> &amp; <font color="Olive">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="Olive">w</font> ) )
 ;<br/>

<a NAME="E4:59"/><b>then </b>
<font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:59"/><b>then </b><i><font color="Green">E48</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">k</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font>;<br/>



<b>assume </b><a NAME="E6:59"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K1">CompactSublatt</a> <font color="Maroon">T</font></span>)</span>
 ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:59_1"/>
<font color="Maroon">k</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>
 ;<br/><a NAME="E2:59_1"/><b>then </b>
<font color="Maroon">k</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_8.html#T1">WAYBEL_8:1</a></i>;<br/><b>then consider </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><br/><a NAME="E4:59_1"/><i><font color="Green">E52</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">k</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> )
 <b>by </b><i><a class="txt" href="waybel17.html#E1:59"><i><font color="Green">E43</font></i></a></i>;<br/><b>consider </b><font color="Maroon">w1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E6:59_1"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">w1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K1">CompactSublatt</a> <font color="Maroon">S</font></span>)</span> &amp; <font color="Maroon">w</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">w1</font> &amp; <font color="Maroon">w1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x</font> )
 <b>by </b><i>, <a class="txt" href="waybel17.html#E2:59"><i><font color="Green">E46</font></i></a>, <a class="ref" href="waybel_8.html#T7">WAYBEL_8:7</a></i>;<br/><a NAME="E7:59_1"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">w</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">k</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font> )
 <b>by </b><i><a class="txt" href="waybel17.html#E2:59"><i><font color="Green">E46</font></i></a>, <a class="ref" href="waybel_3.html#T1">WAYBEL_3:1</a></i>;<br/><b>take </b><font color="Maroon">w1</font> = <font color="Maroon">w1</font>;<br/><b>thus </b><a NAME="E8:59_1"/>
<font color="Maroon">w1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K1">CompactSublatt</a> <font color="Maroon">S</font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel17.html#E3:59"><i><font color="Green">E47</font></i></a></i>;<br/><b>thus </b><a NAME="E9:59_1"/>
<font color="Maroon">w1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="txt" href="waybel17.html#E3:59"><i><font color="Green">E47</font></i></a></i>;<br/><a NAME="E10:59_1"/>
<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">w1</font>
 <b>by </b><i>, <a class="txt" href="waybel17.html#E3:59"><i><font color="Green">E47</font></i></a>, <a class="ref" href="waybel_1.html#D2">WAYBEL_1:def 2</a></i>;<br/><b>hence </b><a NAME="E11:59_1"/>
<font color="Maroon">k</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">w1</font>
 <b>by </b><i><a class="txt" href="waybel17.html#E5:59"><i><font color="Green">E48</font></i></a>, <a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/></div>
<b>end;</b></div>

<b>given </b><font color="Maroon">j</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font><b> such that </b><a NAME="E8:59"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K1">CompactSublatt</a> <font color="Maroon">S</font></span>)</span> &amp; <font color="Maroon">j</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">k</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">j</font> )
 ;<br/>

<a NAME="E9:59"/>
<font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i>, <a class="txt" href="waybel17.html#E1:59"><i><font color="Green">E43</font></i></a>, </i>;<br/>
<a NAME="E10:59"/><b>then </b>
<font color="Maroon">f</font> is <a href="orders_3.html#V5">monotone</a>
 
<b>by </b><i/>;<br/>
<a NAME="E11:59"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">j</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="txt" href="waybel17.html#E2:59"><i><font color="Green">E46</font></i></a>, <a class="ref" href="waybel_1.html#D2">WAYBEL_1:def 2</a></i>;<br/>
<b>hence </b><a NAME="E12:59"/>
<font color="Maroon">k</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="txt" href="waybel17.html#E2:59"><i><font color="Green">E46</font></i></a>, <a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/>


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