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<b>let </b><font color="Maroon" title="c1">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"> for <font color="Olive" title="b1">T</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="waybel_0.html#V24" title="WAYBEL_0:attr.24">up-complete</a> <a href="orders_2.html#NM1" title="ORDERS_2:NM.1">Poset</a><br/> for <font color="Olive" title="b2">f</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Olive" title="b2">f</font> <a href="yellow16.html#R1" title="YELLOW16:pred.1">is_a_retraction_of</a> <font color="Olive" title="b1">T</font>,<font color="Maroon" title="c1">S</font> holds <br/><font color="Olive" title="b2">f</font> <a href="yellow16.html#R2" title="YELLOW16:pred.2">is_an_UPS_retraction_of</a> <font color="Olive" title="b1">T</font>,<font color="Maroon" title="c1">S</font></span><br/><b>let </b><font color="Maroon" title="c2">T</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="waybel_0.html#V24" title="WAYBEL_0:attr.24">up-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">f</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Olive" title="b1">f</font> <a href="yellow16.html#R1" title="YELLOW16:pred.1">is_a_retraction_of</a> <font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c1">S</font> holds <br/><font color="Olive" title="b1">f</font> <a href="yellow16.html#R2" title="YELLOW16:pred.2">is_an_UPS_retraction_of</a> <font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c1">S</font></span><br/><b>let </b><font color="Maroon" title="c3">f</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <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="c2">T</font>,<font color="Maroon" title="c1">S</font> implies <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="c2">T</font>,<font color="Maroon" title="c1">S</font> )</span><br/>





<b>assume </b><a NAME="E1:13"/><i><font color="Green" title="E10">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="c2">T</font>,<font color="Maroon" title="c1">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <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="c2">T</font>,<font color="Maroon" title="c1">S</font></span><br/>

<b>hence </b><a NAME="E2:13"/>
<font color="Maroon" title="c3">f</font> is  <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c1">S</font>
 <b>by </b><i><a class="ref" href="yellow16.html#D1" target="_self" title="YELLOW16:def.1">Def1</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow16.html#D2" target="_self" title="YELLOW16:def.2">YELLOW16:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">g</font> being <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font> st <font color="Maroon" title="c3">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c1">S</font></span><br/>

<a NAME="E3:13"/>
<font color="Maroon" title="c1">S</font> is  <a href="yellow_0.html#V4" title="YELLOW_0:attr.4">full</a> <a href="waybel_0.html#V4" title="WAYBEL_0:attr.4">directed-sups-inheriting</a>  <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of <font color="Maroon" title="c2">T</font>
 
<b>by </b><i><a class="txt" href="#E1:13"><i><font color="Green" title="E10">A1</font></i></a>, <a class="ref" href="yellow16.html#D1" target="_self" title="YELLOW16:def.1">Def1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c4">g</font> =  <a href="yellow_9.html#K1" title="YELLOW_9:func.1">incl</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font> as  <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font> <b>by </b><i><a class="ref" href="waybel21.html#T10" title="WAYBEL21:th.10">WAYBEL21:10</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c4">g</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c1">S</font></span><br/>

<b>thus </b><a NAME="E5:13"/>
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c1">S</font>
 <b>by </b><i><a class="txt" href="#E1:13"><i><font color="Green" title="E10">A1</font></i></a>, <a class="ref" href="yellow16.html#T10" target="_self" title="YELLOW16:th.10">Th10</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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