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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="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> <a href="orders_2.html#NM1" title="ORDERS_2:NM.1">Poset</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">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>  st <font color="Olive" title="b1">S</font> <a href="yellow16.html#R4" title="YELLOW16:pred.4">is_an_UPS_retract_of</a> <font color="Maroon" title="c1">L</font> holds <br/><font color="Olive" title="b1">S</font> is <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a></span><br/><b>let </b><font color="Maroon" title="c2">S</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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">S</font> <a href="yellow16.html#R4" title="YELLOW16:pred.4">is_an_UPS_retract_of</a> <font color="Maroon" title="c1">L</font> implies <font color="Maroon" title="c2">S</font> is <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> )</span><br/>



<b>given </b><font color="Maroon" title="c3">f</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="c2">S</font><b> such that </b><a NAME="E2:28"/><i><font color="Green" title="E22">A1</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="yellow16.html#R2" title="YELLOW16:pred.2">is_an_UPS_retraction_of</a> <font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">S</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow16.html#D4" target="_self" title="YELLOW16:def.4">YELLOW16:def 4</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">S</font> is <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a></span><br/>

<b>consider </b><font color="Maroon" title="c4">h</font> being  <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a> <a href="waybel_1.html#V6" title="WAYBEL_1:attr.6">projection</a> <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><b> such that </b><br/><a NAME="E4:28"/><i><font color="Green" title="E23">A2</font></i>: 
( <font color="Maroon" title="c4">h</font> <a href="yellow16.html#R1" title="YELLOW16:pred.1">is_a_retraction_of</a> <font color="Maroon" title="c1">L</font>, <a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c4">h</font> &amp; <font color="Maroon" title="c2">S</font>, <a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c4">h</font> <a href="waybel_1.html#R5" title="WAYBEL_1:pred.5">are_isomorphic</a>  )
 <b>by </b><i><a class="txt" href="yellow16.html#E2:28"><i><font color="Green" title="E22">A1</font></i></a>, <a class="ref" href="yellow16.html#T21" target="_self" title="YELLOW16:th.21">Th21</a></i>;<br/>
<a NAME="E5:28"/>
<font color="Maroon" title="c4">h</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="waybel_1.html#K2" title="WAYBEL_1:func.2">corestr</a> <font color="Maroon" title="c4">h</font>
 
<b>by </b><i><a class="ref" href="waybel_1.html#T32" title="WAYBEL_1:th.32">WAYBEL_1:32</a></i>;<br/>
<a NAME="E6:28"/><b>then </b>
 <a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c4">h</font> <a href="yellow16.html#R3" title="YELLOW16:pred.3">is_a_retract_of</a> <font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="txt" href="yellow16.html#E4:28"><i><font color="Green" title="E23">A2</font></i></a>, <a class="ref" href="yellow16.html#D3" target="_self" title="YELLOW16:def.3">Def3</a></i>;<br/>
<a NAME="E7:28"/><b>then </b>
(  <a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c4">h</font> is <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> &amp;  <a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c4">h</font>,<font color="Maroon" title="c2">S</font> <a href="waybel_1.html#R5" title="WAYBEL_1:pred.5">are_isomorphic</a>  )
 
<b>by </b><i><a class="txt" href="yellow16.html#E4:28"><i><font color="Green" title="E23">A2</font></i></a>, <a class="ref" href="yellow16.html#T23" target="_self" title="YELLOW16:th.23">Th23</a>, <a class="ref" href="waybel_1.html#T7" title="WAYBEL_1:th.7">WAYBEL_1:7</a></i>;<br/>
<b>hence </b><a NAME="E8:28"/>
<font color="Maroon" title="c2">S</font> is <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a>
 <b>by </b><i><a class="ref" href="waybel20.html#T18" title="WAYBEL20:th.18">WAYBEL20:18</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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