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<b>let </b><font color="Maroon" title="c1">S</font>, <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"> ( <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font> <a href="waybel_1.html#R5" title="WAYBEL_1:pred.5">are_isomorphic</a>  implies ( <font color="Maroon" title="c1">S</font> <a href="yellow16.html#R4" title="YELLOW16:pred.4">is_an_UPS_retract_of</a> <font color="Maroon" title="c2">T</font> &amp; <font color="Maroon" title="c2">T</font> <a href="yellow16.html#R4" title="YELLOW16:pred.4">is_an_UPS_retract_of</a> <font color="Maroon" title="c1">S</font> ) )</span><br/>

<b>assume </b><a NAME="E1:21"/>
<font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font> <a href="waybel_1.html#R5" title="WAYBEL_1:pred.5">are_isomorphic</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">S</font> <a href="yellow16.html#R4" title="YELLOW16:pred.4">is_an_UPS_retract_of</a> <font color="Maroon" title="c2">T</font> &amp; <font color="Maroon" title="c2">T</font> <a href="yellow16.html#R4" title="YELLOW16:pred.4">is_an_UPS_retract_of</a> <font color="Maroon" title="c1">S</font> )</span><br/>

<b>then consider </b><font color="Maroon" title="c3">f</font> being  <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</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>, <font color="Maroon" title="c4">g</font> being  <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</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> such that </b><br/><a NAME="E3:21"/><i><font color="Green" title="E17">A1</font></i>: 
( <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">g</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">T</font> &amp; <font color="Maroon" title="c4">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</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="ref" href="yellow16.html#T18" target="_self" title="YELLOW16:th.18">Th18</a></i>;<br/>
<a NAME="E4:21"/>
( <font color="Maroon" title="c3">f</font> is <a href="waybel_0.html#V23" title="WAYBEL_0:attr.23">isomorphic</a> &amp; <font color="Maroon" title="c4">g</font> is <a href="waybel_0.html#V23" title="WAYBEL_0:attr.23">isomorphic</a> )
 
<b>by </b><i><a class="txt" href="#E3:21"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="yellow16.html#T17" target="_self" title="YELLOW16:th.17">Th17</a></i>;<br/>
<a NAME="E5:21"/><b>then </b>
( <font color="Maroon" title="c3">f</font> is <a href="waybel_0.html#V18" title="WAYBEL_0:attr.18">sups-preserving</a> &amp; <font color="Maroon" title="c4">g</font> is <a href="waybel_0.html#V18" title="WAYBEL_0:attr.18">sups-preserving</a> )
 
<b>by </b><i><a class="ref" href="waybel13.html#T20" title="WAYBEL13:th.20">WAYBEL13:20</a></i>;<br/>
<a NAME="E6:21"/><b>then </b>
( ( for <font color="Olive" title="b1">X</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">S</font>  st not <font color="Olive" title="b1">X</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; <font color="Olive" title="b1">X</font> is <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> holds <br/><font color="Maroon" title="c3">f</font> <a href="waybel_0.html#R4" title="WAYBEL_0:pred.4">preserves_sup_of</a> <font color="Olive" title="b1">X</font> ) &amp; ( for <font color="Olive" title="b1">X</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font>  st not <font color="Olive" title="b1">X</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; <font color="Olive" title="b1">X</font> is <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> holds <br/><font color="Maroon" title="c4">g</font> <a href="waybel_0.html#R4" title="WAYBEL_0:pred.4">preserves_sup_of</a> <font color="Olive" title="b1">X</font> ) )
 
<b>by </b><i><a class="ref" href="waybel_0.html#D33" title="WAYBEL_0:def.33">WAYBEL_0:def 33</a></i>;<br/>
<a NAME="E7:21"/><b>then </b>
( <font color="Maroon" title="c3">f</font> is <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a> &amp; <font color="Maroon" title="c4">g</font> is <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a> )
 
<b>by </b><i><a class="ref" href="waybel_0.html#D37" title="WAYBEL_0:def.37">WAYBEL_0:def 37</a></i>;<br/>
<a NAME="E8:21"/><b>then </b>
( <font color="Maroon" title="c4">g</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> &amp; <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">S</font>,<font color="Maroon" title="c2">T</font> )
 
<b>by </b><i><a class="txt" href="#E3:21"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="yellow16.html#D2" target="_self" title="YELLOW16:def.2">Def2</a></i>;<br/>
<b>hence </b><a NAME="E9:21"/>
( <font color="Maroon" title="c1">S</font> <a href="yellow16.html#R4" title="YELLOW16:pred.4">is_an_UPS_retract_of</a> <font color="Maroon" title="c2">T</font> &amp; <font color="Maroon" title="c2">T</font> <a href="yellow16.html#R4" title="YELLOW16:pred.4">is_an_UPS_retract_of</a> <font color="Maroon" title="c1">S</font> )
 <b>by </b><i><a class="ref" href="yellow16.html#D4" target="_self" title="YELLOW16:def.4">Def4</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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