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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>given </b><font color="Maroon">c<sub>3</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>2</sub></font><b> such that </b><a NAME="E1:27"/><i><font color="Green">E23</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>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow16.html#D4" target="_self">YELLOW16:def 4</a><br/>

<b>consider </b><font color="Maroon">c<sub>4</sub></font> being  <a href="waybel_0.html#V22">directed-sups-preserving</a> <a href="waybel_1.html#V6">projection</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> such that </b><br/><a NAME="E3:27"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="yellow16.html#R1" target="_self">is_a_retraction_of</a> <font color="Maroon">c<sub>1</sub></font>, <a href="yellow_2.html#K1">Image</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font>, <a href="yellow_2.html#K1">Image</a> <font color="Maroon">c<sub>4</sub></font> <a href="waybel_1.html#R5">are_isomorphic</a>  )
 <b>by </b><i><a class="txt" href="yellow16.html#E1:27"><i><font color="Green">E23</font></i></a>, <a class="ref" href="yellow16.html#T21" target="_self">Th21</a></i>;<br/>
<a NAME="E4:27"/>
<font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#R1">=</a>  <a href="waybel_1.html#K2">corestr</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="waybel_1.html#T32">WAYBEL_1:32</a></i>;<br/>
<a NAME="E5:27"/><b>then </b>
 <a href="yellow_2.html#K1">Image</a> <font color="Maroon">c<sub>4</sub></font> <a href="yellow16.html#R3" target="_self">is_a_retract_of</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="yellow16.html#E3:27"><i><font color="Green">E24</font></i></a>, <a class="ref" href="yellow16.html#D3" target="_self">Def3</a></i>;<br/>
<a NAME="E6:27"/><b>then </b>
(  <a href="yellow_2.html#K1">Image</a> <font color="Maroon">c<sub>4</sub></font> is <a href="waybel_0.html#V24">up-complete</a> &amp;  <a href="yellow_2.html#K1">Image</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="waybel_1.html#R5">are_isomorphic</a>  )
 
<b>by </b><i><a class="txt" href="yellow16.html#E3:27"><i><font color="Green">E24</font></i></a>, <a class="ref" href="yellow16.html#T22" target="_self">Th22</a>, <a class="ref" href="waybel_1.html#T7">WAYBEL_1:7</a></i>;<br/>
<b>hence </b><a NAME="E7:27"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="waybel_0.html#V24">up-complete</a>
 <b>by </b><i><a class="ref" href="waybel13.html#T30">WAYBEL13:30</a></i>;<br/>


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