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<b>let </b><font color="Maroon">S</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>, <font color="Maroon">T</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>assume </b><a NAME="E1:21"/>
<font color="Maroon">S</font>,<font color="Maroon">T</font> <a href="waybel_1.html#R5">are_isomorphic</a> 
 ;<br/>

<b>then consider </b><font color="Maroon">f</font> being  <a href="orders_3.html#V5">monotone</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">S</font>,<font color="Maroon">T</font>, <font color="Maroon">g</font> being  <a href="orders_3.html#V5">monotone</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">T</font>,<font color="Maroon">S</font><b> such that </b><br/><a NAME="E3:21"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">g</font> <a href="funct_2.html#R2">=</a>  <a href="grcat_1.html#K7">id</a> <font color="Maroon">T</font> &amp; <font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font> <a href="funct_2.html#R2">=</a>  <a href="grcat_1.html#K7">id</a> <font color="Maroon">S</font> )
 <b>by </b><i/>;<br/>
<a NAME="E4:21"/>
( <font color="Maroon">f</font> is <a href="waybel_0.html#V23">isomorphic</a> &amp; <font color="Maroon">g</font> is <a href="waybel_0.html#V23">isomorphic</a> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:21"/><b>then </b>
( <font color="Maroon">f</font> is <a href="waybel_0.html#V18">sups-preserving</a> &amp; <font color="Maroon">g</font> is <a href="waybel_0.html#V18">sups-preserving</a> )
 
<b>by </b><i><a class="ref" href="waybel13.html#T20">WAYBEL13:20</a></i>;<br/>
<a NAME="E6:21"/><b>then </b>
( ( for <font color="Olive">X</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font>  st not <font color="Olive">X</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Olive">X</font> is <a href="waybel_0.html#V1">directed</a> holds <br/><font color="Maroon">f</font> <a href="waybel_0.html#R4">preserves_sup_of</a> <font color="Olive">X</font> ) &amp; ( for <font color="Olive">X</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>  st not <font color="Olive">X</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Olive">X</font> is <a href="waybel_0.html#V1">directed</a> holds <br/><font color="Maroon">g</font> <a href="waybel_0.html#R4">preserves_sup_of</a> <font color="Olive">X</font> ) )
 
<b>by </b><i><a class="ref" href="waybel_0.html#D33">WAYBEL_0:def 33</a></i>;<br/>
<a NAME="E7:21"/><b>then </b>
( <font color="Maroon">f</font> is <a href="waybel_0.html#V22">directed-sups-preserving</a> &amp; <font color="Maroon">g</font> is <a href="waybel_0.html#V22">directed-sups-preserving</a> )
 
<b>by </b><i><a class="ref" href="waybel_0.html#D37">WAYBEL_0:def 37</a></i>;<br/>
<a NAME="E8:21"/><b>then </b>
( <font color="Maroon">g</font> <a href="yellow16.html#R2" target="_self">is_an_UPS_retraction_of</a> <font color="Maroon">T</font>,<font color="Maroon">S</font> &amp; <font color="Maroon">f</font> <a href="yellow16.html#R2" target="_self">is_an_UPS_retraction_of</a> <font color="Maroon">S</font>,<font color="Maroon">T</font> )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E9:21"/>
( <font color="Maroon">S</font> <a href="yellow16.html#R4" target="_self">is_an_UPS_retract_of</a> <font color="Maroon">T</font> &amp; <font color="Maroon">T</font> <a href="yellow16.html#R4" target="_self">is_an_UPS_retract_of</a> <font color="Maroon">S</font> )
 <b>by </b><i/>;<br/>


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