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<b>let </b><font color="Maroon">L1</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">L2</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">L3</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/><b>let </b><font color="Maroon">f</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">L1</font>,<font color="Maroon">L2</font>;<br/><b>let </b><font color="Maroon">g</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">L2</font>,<font color="Maroon">L3</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:27"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">f</font> is <a href="waybel_0.html#V21">filtered-infs-preserving</a>
 <b>and </b><br/><a NAME="E2:27"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">g</font> is <a href="waybel_0.html#V21">filtered-infs-preserving</a>
 ;<br/>

<b>set </b><font color="Maroon">gf</font> = <font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font>;<br/>
<b>let </b><font color="Maroon">X</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L1</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><b>that </b><br/><a NAME="E3:27"/><i><font color="Green">E37</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> )
 <b>and </b><br/><a NAME="E4:27"/><i><font color="Green">E39</font></i>: 
 <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">X</font>,<font color="Maroon">L1</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D30">WAYBEL_0:def 30</a><br/>

<a NAME="E5:27"/><i><font color="Green">E40</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L1</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>set </b><font color="Maroon">gfX</font> = <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font>;<br/>
<b>set </b><font color="Maroon">fX</font> = <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font>;<br/>
<a NAME="E6:27"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">f</font> <a href="waybel_0.html#R3">preserves_inf_of</a> <font color="Maroon">X</font>
 
<b>by </b><i>, , <a class="ref" href="waybel_0.html#D36">WAYBEL_0:def 36</a></i>;<br/>
<b>consider </b><font color="Maroon">xx</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>;<br/>
<a NAME="E8:27"/>
( <font color="Maroon">xx</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L1</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E9:27"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">xx</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/>
<a NAME="E10:27"/>
<font color="Maroon">f</font> is <a href="orders_3.html#V5">monotone</a>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E11:27"/><b>then </b>
( not <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font> is <a href="waybel_0.html#V2">filtered</a> )
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E12:27"/><b>then </b><i><font color="Green">E44</font></i>: 
<font color="Maroon">g</font> <a href="waybel_0.html#R3">preserves_inf_of</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font>
 
<b>by </b><i>, <a class="ref" href="waybel_0.html#D36">WAYBEL_0:def 36</a></i>;<br/>
<a NAME="E13:27"/><i><font color="Green">E45</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T159">RELAT_1:159</a></i>;<br/>
<a NAME="E14:27"/><i><font color="Green">E46</font></i>: 
 <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font>,<font color="Maroon">L2</font>
 
<b>by </b><i><a class="ref" href="waybel20.html#T1" target="_self">Th1</a>, , <a class="ref" href="waybel_0.html#D30">WAYBEL_0:def 30</a></i>;<br/>
<b>hence </b><a NAME="E15:27"/>
 <a href="yellow_0.html#R2">ex_inf_of</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font>,<font color="Maroon">L3</font>
 <b>by </b><i>, <a class="txt" href="waybel20.html#E1:27"><i><font color="Green">E30</font></i></a>, <a class="ref" href="waybel_0.html#D30">WAYBEL_0:def 30</a></i>;<br/>

<b>thus </b> <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font></span>)</span> = 
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="txt" href="waybel20.html#E1:27"><i><font color="Green">E30</font></i></a>, <a class="txt" href="waybel20.html#E2:27"><i><font color="Green">E31</font></i></a>, <a class="ref" href="waybel_0.html#D30">WAYBEL_0:def 30</a></i>
<br/>.= 
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">X</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="waybel20.html#T1" target="_self">Th1</a>, , <a class="ref" href="waybel_0.html#D30">WAYBEL_0:def 30</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.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>
<b>by </b><i>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
;<br/>


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