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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#R3" title="YELLOW16:pred.3">is_a_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#R3" title="YELLOW16:pred.3">is_a_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="E2:25"/><i><font color="Green" title="E21">A1</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="yellow16.html#R1" title="YELLOW16:pred.1">is_a_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#D3" target="_self" title="YELLOW16:def.3">YELLOW16:def 3</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>reconsider </b><font color="Maroon" title="c4">f</font> = <font color="Maroon" title="c3">f</font> as  <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>by </b><i><a class="txt" href="#E2:25"><i><font color="Green" title="E21">A1</font></i></a>, <a class="ref" href="yellow16.html#T14" target="_self" title="YELLOW16:th.14">Th14</a>, <a class="ref" href="yellow16.html#T16" target="_self" title="YELLOW16:th.16">Th16</a></i>;<br/>
<a NAME="E4:25"/>
(  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c2">S</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c4">f</font> &amp;  <a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c4">f</font> is <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> )
 
<b>by </b><i><a class="txt" href="#E2:25"><i><font color="Green" title="E21">A1</font></i></a>, <a class="ref" href="yellow16.html#T15" target="_self" title="YELLOW16:th.15">Th15</a>, <a class="ref" href="yellow_2.html#T37" title="YELLOW_2:th.37">YELLOW_2:37</a></i>;<br/>
<b>hence </b><a NAME="E5:25"/>
<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="yellow_0.html#T3" title="YELLOW_0:th.3">YELLOW_0:3</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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