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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="E1:75"/>
<font color="Maroon">c<sub>2</sub></font> <a href="waybel_1.html#NR10" target="_self">is_a_projection_operator</a> 
 ;<br/>

<a NAME="E2:75"/><b>then </b><i><font color="Green">E48</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> is <a href="orders_3.html#V5">monotone</a> &amp; <font color="Maroon">c<sub>2</sub></font> is <a href="quantal1.html#V11">idempotent</a> )
 
<b>by </b><i><a class="ref" href="waybel_1.html#D13" target="_self">Def13</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> =  <a href="yellow_2.html#K1">Image</a> <font color="Maroon">c<sub>2</sub></font> as  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#NM1">Poset</a> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> =  <a href="waybel_1.html#K2" target="_self">corestr</a> <font color="Maroon">c<sub>2</sub></font> as   <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> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> =  <a href="waybel_1.html#K3" target="_self">inclusion</a> <font color="Maroon">c<sub>2</sub></font> as   <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> ;<br/>
<b>take </b>
<font color="Maroon">c<sub>3</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>4</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/>

<b>thus </b><a NAME="E6:75"/>
<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#E2:75"><i><font color="Green">E48</font></i></a>, <a class="ref" href="waybel_1.html#T33" target="_self">Th33</a></i>;<br/>

<b>thus </b><a NAME="E7:75"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="funct_2.html#V2">onto</a>
 ;<br/>

<b>thus </b><a NAME="E8:75"/>
( <font color="Maroon">c<sub>5</sub></font> is <a href="orders_3.html#V5">monotone</a> &amp; <font color="Maroon">c<sub>5</sub></font> is <a href="funct_1.html#V2">one-to-one</a> )
 ;<br/>

<b>thus </b><a NAME="E9:75"/>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="waybel_1.html#T35" target="_self">Th35</a></i>;<br/>

<b>thus </b><a NAME="E10:75"/>
 <a href="grcat_1.html#K7">id</a> <font color="Maroon">c<sub>3</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>
 <b>by </b><i><a class="txt" href="waybel_1.html#E2:75"><i><font color="Green">E48</font></i></a>, <a class="ref" href="waybel_1.html#T36" target="_self">Th36</a></i>;<br/>


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