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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="waybel_0.html#V24">up-complete</a> <a href="orders_2.html#NM1">Poset</a>;<br/><b>let </b><font color="Maroon">c<sub>2</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>3</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="E1:18"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="yellow16.html#R1" target="_self">is_a_retraction_of</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as  <a href="waybel_0.html#V22">directed-sups-preserving</a> <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="yellow16.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E3:18"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="yellow16.html#R2" target="_self">is_an_UPS_retraction_of</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="yellow16.html#E1:18"><i><font color="Green">E16</font></i></a>, <a class="ref" href="yellow16.html#T11" target="_self">Th11</a></i>;<br/>
<a NAME="E4:18"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="quantal1.html#V11">idempotent</a>
 
<b>by </b><i><a class="txt" href="yellow16.html#E1:18"><i><font color="Green">E16</font></i></a>, <a class="ref" href="yellow16.html#T14" target="_self">Th14</a></i>;<br/>
<b>then </b><i><font color="Green">E18</font></i>: <font color="Maroon">c<sub>3</sub></font> = 
<font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="quantal1.html#D9">QUANTAL1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="funct_4.html#T2">FUNCT_4:2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="yellow16.html#E3:18"><i><font color="Green">E17</font></i></a>, <a class="ref" href="yellow16.html#T13" target="_self">Th13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="yellow16.html#E1:18"><i><font color="Green">E16</font></i></a>, <a class="ref" href="yellow16.html#D1" target="_self">Def1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font>

;<br/>
<a NAME="E6:18"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is  non <a href="struct_0.html#V3">empty</a> <a href="yellow_0.html#V4">full</a> <a href="waybel_0.html#V4">directed-sups-inheriting</a>  <a href="yellow_0.html#M1">SubRelStr</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="yellow16.html#E1:18"><i><font color="Green">E16</font></i></a>, <a class="ref" href="yellow16.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E7:18"/><b>then </b><i><font color="Green">E20</font></i>: 
 <a href="yellow_9.html#K1">incl</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> is <a href="waybel_0.html#V22">directed-sups-preserving</a>
 
<b>by </b><i><a class="ref" href="waybel21.html#T10">WAYBEL21:10</a></i>;<br/>
<a NAME="E8:18"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="yellow16.html#E6:18"><i><font color="Green">E19</font></i></a>, <a class="ref" href="yellow_0.html#D13">YELLOW_0:def 13</a></i>;<br/>
<a NAME="E9:18"/><b>then </b><i><font color="Green">E21</font></i>: 
 <a href="yellow_9.html#K1">incl</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#R1">=</a>  <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="yellow_9.html#D1">YELLOW_9:def 1</a></i>;<br/>
<b>hence </b><a NAME="E10:18"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="waybel_0.html#V22">directed-sups-preserving</a>
 <b>by </b><i><a class="txt" href="yellow16.html#E5:18"><i><font color="Green">E18</font></i></a>, <a class="txt" href="yellow16.html#E7:18"><i><font color="Green">E20</font></i></a>, <a class="ref" href="waybel20.html#T29">WAYBEL20:29</a></i>;<br/>

<a NAME="E11:18"/>
<font color="Maroon">c<sub>3</sub></font> is  <a href="quantal1.html#V11">idempotent</a> <a href="waybel_0.html#V22">directed-sups-preserving</a> <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>
 
<b>by </b><i><a class="txt" href="yellow16.html#E1:18"><i><font color="Green">E16</font></i></a>, <a class="txt" href="yellow16.html#E5:18"><i><font color="Green">E18</font></i></a>, <a class="txt" href="yellow16.html#E7:18"><i><font color="Green">E20</font></i></a>, <a class="txt" href="yellow16.html#E9:18"><i><font color="Green">E21</font></i></a>, <a class="ref" href="yellow16.html#T14" target="_self">Th14</a>, <a class="ref" href="waybel20.html#T29">WAYBEL20:29</a></i>;<br/>
<b>hence </b><a NAME="E12:18"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="waybel_1.html#V6">projection</a>
 <b>by </b><i><a class="ref" href="waybel_1.html#D13">WAYBEL_1:def 13</a></i>;<br/>


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