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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#NM1">Poset</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E1:79"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="orders_3.html#V5">monotone</a>
 ;<br/>

<b>given </b><font color="Maroon">c<sub>3</sub></font> being  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#NM1">Poset</a>, <font color="Maroon">c<sub>4</sub></font> being   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>5</sub></font> being   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font><b> such that </b><a NAME="E2:79"/><i><font color="Green">E51</font></i>: 
<span class="p1"><a href="waybel_1.html#K1" target="_self">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="waybel_1.html#K1" target="_self">]</a></span> is <a href="waybel_1.html#V3" target="_self">Galois</a>
 <b>and </b><br/><a NAME="E3:79"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<a NAME="E4:79"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font> <a href="yellow_2.html#R1">&lt;=</a>  <a href="grcat_1.html#K7">id</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="grcat_1.html#K7">id</a> <font color="Maroon">c<sub>1</sub></font> <a href="yellow_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="txt" href="waybel_1.html#E2:79"><i><font color="Green">E51</font></i></a>, <a class="ref" href="waybel_1.html#T19" target="_self">Th19</a></i>;<br/>
<a NAME="E5:79"/>
( <font color="Maroon">c<sub>5</sub></font> is <a href="orders_3.html#V5">monotone</a> &amp; <font color="Maroon">c<sub>4</sub></font> is <a href="orders_3.html#V5">monotone</a> )
 
<b>by </b><i><a class="txt" href="waybel_1.html#E2:79"><i><font color="Green">E51</font></i></a>, <a class="ref" href="waybel_1.html#T9" target="_self">Th9</a></i>;<br/>
<a NAME="E6:79"/><b>then </b>
( <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="waybel_1.html#E4:79"><i><font color="Green">E53</font></i></a>, <a class="ref" href="waybel_1.html#T21" target="_self">Th21</a></i>;<br/>
<b>hence </b><a NAME="E7:79"/>
( <font color="Maroon">c<sub>2</sub></font> is <a href="quantal1.html#V11">idempotent</a> &amp; <font color="Maroon">c<sub>2</sub></font> is <a href="orders_3.html#V5">monotone</a> )
 <b>by </b><i><a class="txt" href="waybel_1.html#E1:79"><i><font color="Green">E50</font></i></a>, <a class="txt" href="waybel_1.html#E3:79"><i><font color="Green">E52</font></i></a>, <a class="ref" href="waybel_1.html#T22" target="_self">Th22</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_1.html#D13" target="_self">WAYBEL_1:def 13</a>,<a class="ref" href="waybel_1.html#D14" target="_self">WAYBEL_1:def 14</a><br/>

<b>thus </b><a NAME="E8:79"/>
 <a href="grcat_1.html#K7">id</a> <font color="Maroon">c<sub>1</sub></font> <a href="yellow_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="waybel_1.html#E2:79"><i><font color="Green">E51</font></i></a>, <a class="txt" href="waybel_1.html#E3:79"><i><font color="Green">E52</font></i></a>, <a class="ref" href="waybel_1.html#T19" target="_self">Th19</a></i>;<br/>


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