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<b>let </b><font color="Maroon">S</font> be  non <a href="struct_0.html#V3">empty</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#L1">RelStr</a> ;<br/><b>let </b><font color="Maroon">f</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">S</font>,<font color="Maroon">T</font>;<br/>



<b>assume </b><a NAME="E1:102"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">X</font> being <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font> holds  <font color="Maroon">f</font> <a href="waybel_0.html#R4">preserves_sup_of</a> <font color="Olive">X</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel34.html#D15" target="_self">WAYBEL34:def 15</a><br/>

<b>thus </b><a NAME="E2:102"/>
<font color="Maroon">f</font> is <a href="waybel_0.html#V20">join-preserving</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D35">WAYBEL_0:def 35</a><br/>

<b>thus </b><a NAME="E1:102_1"/>
<font color="Maroon">f</font> <a href="waybel_0.html#R4">preserves_sup_of</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K7">}</a></span>
 <b>by </b><i><a class="ref" href="waybel34.html#T6" target="_self">Th6</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E3:102"/>
<font color="Maroon">f</font> <a href="waybel_0.html#R4">preserves_sup_of</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="waybel34.html#T6" target="_self">Th6</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel34.html#D16" target="_self">WAYBEL34:def 16</a><br/>


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