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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#V3">transitive</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#V3">transitive</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#V3">transitive</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>assume </b><b>that </b><br/><a NAME="E1:24"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">f</font> is <a href="orders_3.html#V5">monotone</a> &amp; <font color="Maroon">f</font> is <a href="waybel_0.html#V22">directed-sups-preserving</a> )
 <b>and </b><br/><a NAME="E2:24"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">L2</font> is  <a href="yellow_0.html#V4">full</a> <a href="waybel_0.html#V4">directed-sups-inheriting</a>  <a href="yellow_0.html#M1">SubRelStr</a> of <font color="Maroon">L3</font>
 <b>and </b><br/><a NAME="E3:24"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">L3</font> is <a href="lattice3.html#V3">complete</a>
 ;<br/>

<a NAME="E4:24"/>
( not the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L2</font> is <a href="xboole_0.html#V1">empty</a> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L2</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L3</font> )
 
<b>by </b><i>, <a class="ref" href="yellow_0.html#D13">YELLOW_0:def 13</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">g</font> = <font color="Maroon">f</font> as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">L1</font>,<font color="Maroon">L3</font> <b>by </b><i><a class="ref" href="funct_2.html#T9">FUNCT_2:9</a></i>;<br/>
<b>take </b>
<font color="Maroon">g</font>
;<br/>

<b>thus </b><a NAME="E6:24"/>
<font color="Maroon">f</font> <a href="funct_2.html#R1">=</a> <font color="Maroon">g</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">X</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L1</font>;<br/><b>assume </b><a NAME="E1:24_1"/><i><font color="Green">E36</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#V1">directed</a> )
 ;<br/><b>then consider </b><font color="Maroon">d</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:24_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><a NAME="E4:24_1"/>
( not the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L2</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L1</font> )
 <b>by </b><i/>;<br/><a NAME="E5:24_1"/><b>then </b>
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E6:24_1"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">d</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_1.html#D12">FUNCT_1:def 12</a></i>;<br/><a NAME="E7:24_1"/><b>then </b><i><font color="Green">E50</font></i>: 
( 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#V1">directed</a> )
 <b>by </b><i>, , <a class="ref" href="yellow_2.html#T17">YELLOW_2:17</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:24_1_1"/><i><font color="Green">E56</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">X</font>,<font color="Maroon">L1</font>
 ;<br/><a NAME="E2:24_1_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">f</font> <a href="waybel_0.html#R4">preserves_sup_of</a> <font color="Maroon">X</font>
 <b>by </b><i>, , <a class="ref" href="waybel_0.html#D37">WAYBEL_0:def 37</a></i>;<br/><b>thus </b><a NAME="E3:24_1_1"/>
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font>,<font color="Maroon">L3</font>
 <b>by </b><i>, <a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/><b>hence </b> <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font></span>)</span> = 
 <a href="yellow_0.html#NK3" target="_self">sup</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="waybel_0.html#T7">WAYBEL_0:7</a></i>
<br/>.= 
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">X</font></span>)</span>
<b>by </b><i><a class="ref" href="waybel13.html#T1" target="_self">Th1</a>, , <a class="ref" href="waybel_0.html#D31">WAYBEL_0:def 31</a></i>
;<br/><br/></div><b>end;</b></div><b>hence </b><a NAME="E9:24_1"/>
<font color="Maroon">g</font> <a href="waybel_0.html#R4">preserves_sup_of</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="waybel_0.html#D31">WAYBEL_0:def 31</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:24"/>
<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/>


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