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<div class="add">

<b>let </b><font color="Maroon">S</font> be  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#V2">reflexive</a> <a href="orders_2.html#V4">antisymmetric</a>  <a href="orders_2.html#L1">RelStr</a> , <font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#V2">reflexive</a> <a href="orders_2.html#V4">antisymmetric</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/>

<a NAME="E1:47"/><i><font color="Green">E36</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="waybel27.html#K4" target="_self">UPS</a> <span class="p2">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">S</font></span>)</span>,<span class="p2">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">T</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel27.html#K2" target="_self">UPS</a> <font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E2:47"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel27.html#K2" target="_self">UPS</a> <font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="waybel27.html#K4" target="_self">UPS</a> <span class="p2">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">S</font></span>)</span>,<span class="p2">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">T</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:47_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel27.html#K2" target="_self">UPS</a> <font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">x</font> as  <a href="waybel_0.html#V22">directed-sups-preserving</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">S</font>,<font color="Maroon">T</font> <b>by </b><i/>;<br/>
<span class="p1">(<span class="default"><a href="waybel27.html#K4" target="_self">UPS</a> <span class="p2">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">S</font></span>)</span>,<span class="p2">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">T</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">f</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">T</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">S</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">S</font></span>)</span>
<b>by </b><i><a class="ref" href="grcat_1.html#T12">GRCAT_1:12</a></i>
;<br/>
<b>hence </b><a NAME="E4:47_1"/>
<span class="p1">(<span class="default"><a href="waybel27.html#K4" target="_self">UPS</a> <span class="p2">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">S</font></span>)</span>,<span class="p2">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">T</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="grcat_1.html#T12">GRCAT_1:12</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:47"/><b>then </b>
 <a href="waybel27.html#K4" target="_self">UPS</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">S</font></span>)</span>,<span class="p1">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">T</font></span>)</span> <a href="funct_2.html#R2">=</a>  <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel27.html#K2" target="_self">UPS</a> <font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T34">FUNCT_1:34</a></i>;<br/>
<b>hence </b><a NAME="E4:47"/>
 <a href="waybel27.html#K4" target="_self">UPS</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">S</font></span>)</span>,<span class="p1">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">T</font></span>)</span> <a href="funct_2.html#R2">=</a>  <a href="grcat_1.html#K7">id</a> <span class="p1">(<span class="default"><a href="waybel27.html#K2" target="_self">UPS</a> <font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span>
 ;<br/>


</div>
