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<b>let </b><font color="Maroon" title="c1">L</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="orders_2.html#NM1" title="ORDERS_2:NM.1">Poset</a>;<br/><b>let </b><font color="Maroon" title="c2">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/>



<b>assume </b><a NAME="E1:79"/><i><font color="Green" title="E49">A1</font></i>: 
<font color="Maroon" title="c2">f</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a>
 ;<br/>

<b>given </b><font color="Maroon" title="c3">S</font> being  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="orders_2.html#NM1" title="ORDERS_2:NM.1">Poset</a>, <font color="Maroon" title="c4">g</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c1">L</font>, <font color="Maroon" title="c5">d</font> being   <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="c3">S</font><b> such that </b><a NAME="E3:79"/><i><font color="Green" title="E50">A2</font></i>: 
<span class="p1"><a href="waybel_1.html#K1" title="WAYBEL_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c5">d</font></span><a href="waybel_1.html#K1" title="WAYBEL_1:func.1">]</a></span> is <a href="waybel_1.html#V3" title="WAYBEL_1:attr.3">Galois</a>
 <b>and </b><br/><a NAME="E4:79"/><i><font color="Green" title="E51">A3</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c4">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">d</font>
 ;<br/>

<a NAME="E5:79"/><i><font color="Green" title="E52">A4</font></i>: 
( <font color="Maroon" title="c5">d</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">g</font> <a href="yellow_2.html#R1" title="YELLOW_2:pred.1">&lt;=</a>  <a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c3">S</font> &amp;  <a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c1">L</font> <a href="yellow_2.html#R1" title="YELLOW_2:pred.1">&lt;=</a> <font color="Maroon" title="c4">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">d</font> )
 
<b>by </b><i><a class="txt" href="waybel_1.html#E3:79"><i><font color="Green" title="E50">A2</font></i></a>, <a class="ref" href="waybel_1.html#T19" target="_self" title="WAYBEL_1:th.19">Th19</a></i>;<br/>
<a NAME="E6:79"/>
( <font color="Maroon" title="c5">d</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a> &amp; <font color="Maroon" title="c4">g</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a> )
 
<b>by </b><i><a class="txt" href="waybel_1.html#E3:79"><i><font color="Green" title="E50">A2</font></i></a>, <a class="ref" href="waybel_1.html#T9" target="_self" title="WAYBEL_1:th.9">Th9</a></i>;<br/>
<a NAME="E7:79"/><b>then </b>
( <font color="Maroon" title="c5">d</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">d</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">d</font> &amp; <font color="Maroon" title="c4">g</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">d</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">g</font> )
 
<b>by </b><i><a class="txt" href="waybel_1.html#E5:79"><i><font color="Green" title="E52">A4</font></i></a>, <a class="ref" href="waybel_1.html#T21" target="_self" title="WAYBEL_1:th.21">Th21</a></i>;<br/>
<b>hence </b><a NAME="E8:79"/>
( <font color="Maroon" title="c2">f</font> is <a href="quantal1.html#V11" title="QUANTAL1:attr.11">idempotent</a> &amp; <font color="Maroon" title="c2">f</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a> )
 <b>by </b><i><a class="txt" href="waybel_1.html#E1:79"><i><font color="Green" title="E49">A1</font></i></a>, <a class="txt" href="waybel_1.html#E4:79"><i><font color="Green" title="E51">A3</font></i></a>, <a class="ref" href="waybel_1.html#T22" target="_self" title="WAYBEL_1:th.22">Th22</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_1.html#D13" target="_self" title="WAYBEL_1:def.13">WAYBEL_1:def 13</a>,<a class="ref" href="waybel_1.html#D14" target="_self" title="WAYBEL_1:def.14">WAYBEL_1:def 14</a><br/>

<b>thus </b><a NAME="E9:79"/>
 <a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c1">L</font> <a href="yellow_2.html#R1" title="YELLOW_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">f</font>
 <b>by </b><i><a class="txt" href="waybel_1.html#E3:79"><i><font color="Green" title="E50">A2</font></i></a>, <a class="txt" href="waybel_1.html#E4:79"><i><font color="Green" title="E51">A3</font></i></a>, <a class="ref" href="waybel_1.html#T19" target="_self" title="WAYBEL_1:th.19">Th19</a></i>;<br/>


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