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<b>consider </b><font color="Maroon" title="c2">h</font> being  <a href="waybel_1.html#V7" title="WAYBEL_1:attr.7">closure</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c1">L</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>$1 is  <a href="waybel_1.html#V7" title="WAYBEL_1:attr.7">closure</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c1">L</font>;<br/>
<a NAME="E2:24_1_1"/><i><font color="Green" title="E10">A1</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">h</font>]
 
;<br/>
<a NAME="E3:24_1_1"/>
<font color="Maroon" title="c2">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T12" title="FUNCT_2:th.12">FUNCT_2:12</a></i>;<br/>
<a NAME="E4:24_1_1"/><b>then </b><i><font color="Green" title="E11">A2</font></i>: 
<font color="Maroon" title="c2">h</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="yellow_1.html#K7" title="YELLOW_1:func.7">MonMaps</a> <font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c1">L</font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_1.html#D6" title="YELLOW_1:def.6">YELLOW_1:def 6</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c3">S</font> being  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a> <a href="yellow_0.html#V4" title="YELLOW_0:attr.4">full</a>  <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of  <a href="yellow_1.html#K7" title="YELLOW_1:func.7">MonMaps</a> <font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E6:24_1_1"/><i><font color="Green" title="E12">A3</font></i>: 
for <font color="Olive" title="b1">f</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K7" title="YELLOW_1:func.7">MonMaps</a> <font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c1">L</font></span>)</span> holds <br/> ( <font color="Olive" title="b1">f</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">S</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">f</font>] )
 <b>from </b><i><a class="ref" href="waybel10.html#S1" target="_self" title="WAYBEL10:sch.1">WAYBEL10:sch 1</a>(<a class="txt" href="waybel10.html#E2:24_1_1"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="waybel10.html#E4:24_1_1"><i><font color="Green" title="E11">A2</font></i></a>);<br/></i>
<b>take </b>
<font color="Maroon" title="c3">S</font>
;<br/>

<b>let </b><font color="Maroon" title="c4">f</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c1">L</font>;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:24_1_1_1"/><i><font color="Green" title="E13">A4</font></i>: 
<font color="Maroon" title="c4">f</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">S</font>
 ;<br/><a NAME="E2:24_1_1_1"/><b>then </b>
<font color="Maroon" title="c4">f</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K7" title="YELLOW_1:func.7">MonMaps</a> <font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c1">L</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_0.html#T59" title="YELLOW_0:th.59">YELLOW_0:59</a></i>;<br/><b>hence </b><a NAME="E3:24_1_1_1"/>
<font color="Maroon" title="c4">f</font> is <a href="waybel_1.html#V7" title="WAYBEL_1:attr.7">closure</a>
 <b>by </b><i><a class="txt" href="waybel10.html#E6:24_1_1"><i><font color="Green" title="E12">A3</font></i></a>, <a class="txt" href="waybel10.html#E1:24_1_1_1"><i><font color="Green" title="E13">A4</font></i></a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E8:24_1_1"/><i><font color="Green" title="E13">A5</font></i>: 
<font color="Maroon" title="c4">f</font> is <a href="waybel_1.html#V7" title="WAYBEL_1:attr.7">closure</a>
 ;<br/>

<a NAME="E9:24_1_1"/><b>then </b>
<font color="Maroon" title="c4">f</font> is  <a href="waybel_1.html#V7" title="WAYBEL_1:attr.7">closure</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c1">L</font>
 
;<br/>
<a NAME="E10:24_1_1"/><b>then </b>
( <font color="Maroon" title="c4">f</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a> &amp; <font color="Maroon" title="c4">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#T12" title="FUNCT_2:th.12">FUNCT_2:12</a></i>;<br/>
<a NAME="E11:24_1_1"/><b>then </b>
<font color="Maroon" title="c4">f</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="yellow_1.html#K7" title="YELLOW_1:func.7">MonMaps</a> <font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c1">L</font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_1.html#D6" title="YELLOW_1:def.6">YELLOW_1:def 6</a></i>;<br/>
<b>hence </b><a NAME="E12:24_1_1"/>
<font color="Maroon" title="c4">f</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">S</font>
 <b>by </b><i><a class="txt" href="waybel10.html#E6:24_1_1"><i><font color="Green" title="E12">A3</font></i></a>, <a class="txt" href="waybel10.html#E8:24_1_1"><i><font color="Green" title="E13">A5</font></i></a></i>;<br/>


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