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<b>let </b><font color="Maroon">T</font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/><b>let </b><font color="Maroon">S</font> be  non <a href="struct_0.html#V3">empty</a> <a href="yellow_0.html#V4">full</a> <a href="waybel_0.html#V3">filtered-infs-inheriting</a>  <a href="yellow_0.html#M1">SubRelStr</a> of <font color="Maroon">T</font>;<br/>



<b>set </b><font color="Maroon">f</font> =  <a href="yellow_9.html#K1">incl</a> <font color="Maroon">S</font>,<font color="Maroon">T</font>;<br/>
<b>let </b><font color="Maroon">X</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D36">WAYBEL_0:def 36</a><br/>

<b>assume </b><a NAME="E1:14"/><i><font color="Green">E26</font></i>: 
( not <font color="Maroon">X</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">X</font> is <a href="waybel_0.html#V2">filtered</a> &amp;  <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">X</font>,<font color="Maroon">S</font> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D30">WAYBEL_0:def 30</a><br/>

<b>thus </b><a NAME="E2:14"/>
 <a href="yellow_0.html#R2">ex_inf_of</a> <span class="p1">(<span class="default"><a href="yellow_9.html#K1">incl</a> <font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font>,<font color="Maroon">T</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/>

<a NAME="E3:14"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D13">YELLOW_0:def 13</a></i>;<br/>
<a NAME="E4:14"/><b>then </b>
 <a href="yellow_9.html#K1">incl</a> <font color="Maroon">S</font>,<font color="Maroon">T</font> <a href="hidden.html#R1">=</a>  <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="yellow_9.html#D1">YELLOW_9:def 1</a></i>;<br/>
<a NAME="E5:14"/><b>then </b>
( <span class="p1">(<span class="default"><a href="yellow_9.html#K1">incl</a> <font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> &amp;  <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">X</font>,<font color="Maroon">T</font> &amp; <span class="p1">(<span class="default"><a href="yellow_9.html#K1">incl</a> <font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="yellow_6.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">X</font> )
 
<b>by </b><i><a class="ref" href="funct_1.html#T35">FUNCT_1:35</a>, <a class="ref" href="funct_1.html#T162">FUNCT_1:162</a>, <a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/>
<b>hence </b><a NAME="E6:14"/>
 <a href="yellow_0.html#K2">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow_9.html#K1">incl</a> <font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font></span>)</span>,<font color="Maroon">T</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="yellow_9.html#K1">incl</a> <font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">X</font>,<font color="Maroon">S</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="waybel_0.html#T6">WAYBEL_0:6</a></i>;<br/>


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