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<b>let </b><font color="Maroon" title="c1">L</font> be  <a href="yellow_0.html#V1" title="YELLOW_0:attr.1">lower-bounded</a> <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a> <a href="waybel_0.html#NM7" title="WAYBEL_0:NM.7">sup-Semilattice</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">B</font> being <a href="waybel23.html#V6" title="WAYBEL23:attr.6">with_bottom</a>  <a href="waybel23.html#M1" title="WAYBEL23:mode.1">CLbasis</a> of <font color="Maroon" title="c1">L</font> holds   <a href="waybel23.html#K5" title="WAYBEL23:func.5">baseMap</a> <font color="Olive" title="b1">B</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a></span><br/><b>let </b><font color="Maroon" title="c2">B</font> be  <a href="waybel23.html#V6" title="WAYBEL23:attr.6">with_bottom</a>  <a href="waybel23.html#M1" title="WAYBEL23:mode.1">CLbasis</a> of <font color="Maroon" title="c1">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="waybel23.html#K5" title="WAYBEL23:func.5">baseMap</a> <font color="Maroon" title="c2">B</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a></span><br/>



<b>set </b><font color="Maroon" title="c3">f</font> =  <a href="waybel23.html#K5" title="WAYBEL23:func.5">baseMap</a> <font color="Maroon" title="c2">B</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">y</font> implies <span class="p1">(<span class="default"><a href="waybel23.html#K5" title="WAYBEL23:func.5">baseMap</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><a href="waybel23.html#K5" title="WAYBEL23:func.5">baseMap</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">y</font> )</span><br/><b>assume </b><a NAME="E1:96_1"/><i><font color="Green" title="E54">A1</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="waybel23.html#K5" title="WAYBEL23:func.5">baseMap</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><a href="waybel23.html#K5" title="WAYBEL23:func.5">baseMap</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">y</font></span><br/><a NAME="E2:96_1"/><i><font color="Green" title="E55">A2</font></i>: 
<span class="p1">(<span class="default"><a href="waybel23.html#K5" title="WAYBEL23:func.5">baseMap</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">B</font>
 <b>by </b><i><a class="ref" href="waybel23.html#D12" target="_self" title="WAYBEL23:def.12">Def12</a></i>;<br/><a NAME="E3:96_1"/><i><font color="Green" title="E56">A3</font></i>: 
<span class="p1">(<span class="default"><a href="waybel23.html#K5" title="WAYBEL23:func.5">baseMap</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c5">y</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">B</font>
 <b>by </b><i><a class="ref" href="waybel23.html#D12" target="_self" title="WAYBEL23:def.12">Def12</a></i>;<br/><a NAME="E4:96_1"/>
 <a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c4">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c5">y</font>
 <b>by </b><i><a class="txt" href="waybel23.html#E1:96_1"><i><font color="Green" title="E54">A1</font></i></a>, <a class="ref" href="waybel_3.html#T12" title="WAYBEL_3:th.12">WAYBEL_3:12</a></i>;<br/><a NAME="E5:96_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="waybel23.html#K5" title="WAYBEL23:func.5">baseMap</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="waybel23.html#K5" title="WAYBEL23:func.5">baseMap</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">y</font>
 <b>by </b><i><a class="txt" href="waybel23.html#E2:96_1"><i><font color="Green" title="E55">A2</font></i></a>, <a class="txt" href="waybel23.html#E3:96_1"><i><font color="Green" title="E56">A3</font></i></a>, <a class="ref" href="xboole_1.html#T26" title="XBOOLE_1:th.26">XBOOLE_1:26</a></i>;<br/><b>hence </b><a NAME="E6:96_1"/>
<span class="p1">(<span class="default"><a href="waybel23.html#K5" title="WAYBEL23:func.5">baseMap</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><a href="waybel23.html#K5" title="WAYBEL23:func.5">baseMap</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">y</font>
 <b>by </b><i><a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:96"/>
 <a href="waybel23.html#K5" title="WAYBEL23:func.5">baseMap</a> <font color="Maroon" title="c2">B</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a>
 <b>by </b><i><a class="ref" href="waybel_1.html#D2" title="WAYBEL_1:def.2">WAYBEL_1:def 2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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