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<b>let </b><font color="Maroon" title="c1">S</font>, <font color="Maroon" title="c2">T</font> be  <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> <a href="waybel11.html#V4" title="WAYBEL11:attr.4">Scott</a> <a href="waybel_9.html#NM1" title="WAYBEL_9:NM.1">TopLattice</a>;<br/><b>let </b><font color="Maroon" title="c3">f</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font>;<br/>



<b>assume </b><a NAME="E1:59"/><i><font color="Green" title="E41">A1</font></i>: 
( <font color="Maroon" title="c1">S</font> is <a href="waybel_8.html#V2" title="WAYBEL_8:attr.2">algebraic</a> &amp; <font color="Maroon" title="c2">T</font> is <a href="waybel_8.html#V2" title="WAYBEL_8:attr.2">algebraic</a> )
 ;<br/>

<a NAME="E2:59"/><b>then </b><i><font color="Green" title="E42">A2</font></i>: 
( <font color="Maroon" title="c1">S</font> is <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a> &amp; <font color="Maroon" title="c2">T</font> is <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a> )
 
<b>by </b><i><a class="ref" href="waybel_8.html#T7" title="WAYBEL_8:th.7">WAYBEL_8:7</a></i>;<br/>
<b>assume </b><a NAME="E3:59"/><i><font color="Green" title="E43">A3</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font><br/> for <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">T</font> holds <br/> ( <font color="Olive" title="b2">y</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> iff  ex <font color="Olive" title="b3">w</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> st <br/>( <font color="Olive" title="b3">w</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b2">y</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">w</font> ) )
 ;<br/>

<a NAME="E4:59"/><b>then </b>
<font color="Maroon" title="c3">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<b>by </b><i><a class="txt" href="waybel17.html#E2:59"><i><font color="Green" title="E42">A2</font></i></a>, <a class="ref" href="waybel17.html#T23" target="_self" title="WAYBEL17:th.23">Th23</a></i>;<br/>
<a NAME="E5:59"/><b>then </b><i><font color="Green" title="E44">A4</font></i>: 
<font color="Maroon" title="c3">f</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a>
 
<b>by </b><i><a class="ref" href="waybel17.html#T4" target="_self" title="WAYBEL17:th.4">Th4</a></i>;<br/>
<b>let </b><font color="Maroon" title="c4">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font>;<br/><b>let </b><font color="Maroon" title="c5">k</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">T</font>;<br/>



<b>assume </b><a NAME="E6:59"/><i><font color="Green" title="E45">A5</font></i>: 
<font color="Maroon" title="c5">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c2">T</font></span>)</span>
 ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:59_1"/>
<font color="Maroon" title="c5">k</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font>
 ;<br/><a NAME="E2:59_1"/><b>then </b>
<font color="Maroon" title="c5">k</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font>
 <b>by </b><i><a class="txt" href="waybel17.html#E6:59"><i><font color="Green" title="E45">A5</font></i></a>, <a class="ref" href="waybel_8.html#T1" title="WAYBEL_8:th.1">WAYBEL_8:1</a></i>;<br/><b>then consider </b><font color="Maroon" title="c6">w</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font><b> such that </b><br/><a NAME="E4:59_1"/><i><font color="Green" title="E46">A6</font></i>: 
( <font color="Maroon" title="c6">w</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Maroon" title="c5">k</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">w</font> )
 <b>by </b><i><a class="txt" href="waybel17.html#E3:59"><i><font color="Green" title="E43">A3</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c7">w1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font><b> such that </b><br/><a NAME="E6:59_1"/><i><font color="Green" title="E47">A7</font></i>: 
( <font color="Maroon" title="c7">w1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c1">S</font></span>)</span> &amp; <font color="Maroon" title="c6">w</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c7">w1</font> &amp; <font color="Maroon" title="c7">w1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">x</font> )
 <b>by </b><i><a class="txt" href="waybel17.html#E1:59"><i><font color="Green" title="E41">A1</font></i></a>, <a class="txt" href="waybel17.html#E4:59_1"><i><font color="Green" title="E46">A6</font></i></a>, <a class="ref" href="waybel_8.html#T7" title="WAYBEL_8:th.7">WAYBEL_8:7</a></i>;<br/><a NAME="E7:59_1"/><i><font color="Green" title="E48">A8</font></i>: 
( <font color="Maroon" title="c6">w</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Maroon" title="c5">k</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">w</font> )
 <b>by </b><i><a class="txt" href="waybel17.html#E4:59_1"><i><font color="Green" title="E46">A6</font></i></a>, <a class="ref" href="waybel_3.html#T1" title="WAYBEL_3:th.1">WAYBEL_3:1</a></i>;<br/><b>take </b><font color="Maroon" title="c8">w1</font> = <font color="Maroon" title="c7">w1</font>;<br/><b>thus </b><a NAME="E8:59_1"/>
<font color="Maroon" title="c8">w1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c1">S</font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel17.html#E6:59_1"><i><font color="Green" title="E47">A7</font></i></a></i>;<br/><b>thus </b><a NAME="E9:59_1"/>
<font color="Maroon" title="c8">w1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">x</font>
 <b>by </b><i><a class="txt" href="waybel17.html#E6:59_1"><i><font color="Green" title="E47">A7</font></i></a></i>;<br/><a NAME="E10:59_1"/>
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">w</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">w1</font>
 <b>by </b><i><a class="txt" href="waybel17.html#E5:59"><i><font color="Green" title="E44">A4</font></i></a>, <a class="txt" href="waybel17.html#E6:59_1"><i><font color="Green" title="E47">A7</font></i></a>, <a class="ref" href="waybel_1.html#D2" title="WAYBEL_1:def.2">WAYBEL_1:def 2</a></i>;<br/><b>hence </b><a NAME="E11:59_1"/>
<font color="Maroon" title="c5">k</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">w1</font>
 <b>by </b><i><a class="txt" href="waybel17.html#E7:59_1"><i><font color="Green" title="E48">A8</font></i></a>, <a class="ref" href="orders_2.html#T26" title="ORDERS_2:th.26">ORDERS_2:26</a></i>;<br/></div>
<b>end;</b></div>

<b>given </b><font color="Maroon" title="c6">j</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font><b> such that </b><a NAME="E9:59"/><i><font color="Green" title="E46">A9</font></i>: 
( <font color="Maroon" title="c6">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c1">S</font></span>)</span> &amp; <font color="Maroon" title="c6">j</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Maroon" title="c5">k</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">j</font> )
 ;<br/>

<a NAME="E10:59"/>
<font color="Maroon" title="c3">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<b>by </b><i><a class="txt" href="waybel17.html#E2:59"><i><font color="Green" title="E42">A2</font></i></a>, <a class="txt" href="waybel17.html#E3:59"><i><font color="Green" title="E43">A3</font></i></a>, <a class="txt" href="waybel17.html#E39"><i><font color="Green" title="E35">Lm15</font></i></a></i>;<br/>
<a NAME="E11:59"/><b>then </b>
<font color="Maroon" title="c3">f</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a>
 
<b>by </b><i><a class="ref" href="waybel17.html#T4" target="_self" title="WAYBEL17:th.4">Th4</a></i>;<br/>
<a NAME="E12:59"/><b>then </b>
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">j</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font>
 
<b>by </b><i><a class="txt" href="waybel17.html#E9:59"><i><font color="Green" title="E46">A9</font></i></a>, <a class="ref" href="waybel_1.html#D2" title="WAYBEL_1:def.2">WAYBEL_1:def 2</a></i>;<br/>
<b>hence </b><a NAME="E13:59"/>
<font color="Maroon" title="c5">k</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font>
 <b>by </b><i><a class="txt" href="waybel17.html#E9:59"><i><font color="Green" title="E46">A9</font></i></a>, <a class="ref" href="orders_2.html#T26" title="ORDERS_2:th.26">ORDERS_2:26</a></i>;<br/>


</div>
