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<b>let </b><font color="Maroon">L</font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel_3.html#V3">continuous</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/><b>let </b><font color="Maroon">p</font> be  <a href="waybel_1.html#V6">projection</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">L</font>,<font color="Maroon">L</font>;<br/>



<b>assume </b><a NAME="E1:19"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">p</font> is <a href="waybel_0.html#V22">directed-sups-preserving</a>
 ;<br/>

<b>reconsider </b><font color="Maroon">Lk</font> = <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Maroon">p</font> <a href="funct_2.html#K3">.</a> <font color="Olive">k</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">k</font> </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> <b>by </b><i><a class="ref" href="waybel_1.html#T46">WAYBEL_1:46</a></i>;<br/>
<a NAME="E3:19"/><i><font color="Green">E34</font></i>: 
 <a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">Lk</font> is <a href="yellow_0.html#V7">infs-inheriting</a>
 
<b>by </b><i><a class="ref" href="waybel_1.html#T53">WAYBEL_1:53</a></i>;<br/>
<a NAME="E4:19"/>
 <a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">Lk</font> is <a href="waybel_0.html#V4">directed-sups-inheriting</a>
 
<b>by </b><i>, <a class="ref" href="waybel_1.html#T55">WAYBEL_1:55</a></i>;<br/>
<a NAME="E5:19"/><b>then </b><i><font color="Green">E45</font></i>: 
 <a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">Lk</font> is  <a href="waybel_3.html#V3">continuous</a> <a href="waybel_0.html#NM8">LATTICE</a>
 
<b>by </b><i>, <a class="ref" href="waybel_5.html#T28">WAYBEL_5:28</a></i>;<br/>
<a NAME="E6:19"/><i><font color="Green">E46</font></i>: 
 <a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">Lk</font> is <a href="lattice3.html#V3">complete</a>
 
<b>by </b><i>, <a class="ref" href="yellow_2.html#T32">YELLOW_2:32</a></i>;<br/>
<b>reconsider </b><font color="Maroon">pk</font> = <font color="Maroon">p</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">Lk</font> as   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">Lk</font></span>)</span>,<span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">Lk</font></span>)</span> <b>by </b><i><a class="ref" href="waybel_1.html#T49">WAYBEL_1:49</a></i>;<br/>
<a NAME="E8:19"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">pk</font> is <a href="waybel_1.html#V8">kernel</a>
 
<b>by </b><i><a class="ref" href="waybel_1.html#T51">WAYBEL_1:51</a></i>;<br/>
<a NAME="E9:19"/><i><font color="Green">E52</font></i>: 
 <a href="yellow_0.html#K5">subrelstr</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">pk</font></span>)</span> is  <a href="yellow_0.html#V4">full</a>  <a href="yellow_0.html#M1">SubRelStr</a> of <font color="Maroon">L</font>
 
<b>by </b><i/>;<br/>
<i><font color="Green">E53</font></i>: the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">p</font></span>)</span></span>)</span> = 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">p</font>
<b>by </b><i><a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>
<br/>.= 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">pk</font>
<b>by </b><i><a class="ref" href="waybel_1.html#T47">WAYBEL_1:47</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">pk</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>
;<br/>
<a NAME="E11:19"/><i><font color="Green">E55</font></i>: 
 <a href="yellow_2.html#K1">Image</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="yellow_2.html#K1">Image</a> <font color="Maroon">pk</font>
 
<b>by </b><i>, , <a class="ref" href="yellow_0.html#T58">YELLOW_0:58</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2"><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 <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">Lk</font></span>)</span>;<br/><b>assume </b><a NAME="E1:19_2"/><i><font color="Green">E56</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>reconsider </b><font color="Maroon">X'</font> = <font color="Maroon">X</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> <b>by </b><i><a class="ref" href="waybel13.html#T3">WAYBEL13:3</a></i>;<br/><b>reconsider </b><font color="Maroon">X'</font> = <font color="Maroon">X'</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V1">directed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> <b>by </b><i>, <a class="ref" href="yellow_2.html#T7">YELLOW_2:7</a></i>;<br/><a NAME="E4:19_2"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">p</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/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:19_2_1"/>
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">X</font>, <a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">Lk</font>
 ;<br/><a NAME="E2:19_2_1"/>
 <a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">Lk</font> is <a href="yellow_0.html#V7">infs-inheriting</a>
 <b>by </b><i><a class="ref" href="waybel_1.html#T53">WAYBEL_1:53</a></i>;<br/><a NAME="E3:19_2_1"/><b>then </b>
 <a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">Lk</font> is  <a href="lattice3.html#V3">complete</a> <a href="waybel_0.html#NM8">LATTICE</a>
 <b>by </b><i><a class="ref" href="yellow_2.html#T32">YELLOW_2:32</a></i>;<br/><b>hence </b><a NAME="E4:19_2_1"/>
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">pk</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font>, <a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">Lk</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/><a NAME="E5:19_2_1"/><i><font color="Green">E69</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">pk</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">Lk</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E6:19_2_1"/><i><font color="Green">E70</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">X</font>,<font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/><a NAME="E7:19_2_1"/><i><font color="Green">E83</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">p</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font>,<font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/><a NAME="E8:19_2_1"/>
<font color="Maroon">pk</font> is <a href="waybel_1.html#V6">projection</a>
 <b>by </b><i>, <a class="ref" href="waybel_1.html#D15">WAYBEL_1:def 15</a></i>;<br/><a NAME="E9:19_2_1"/><b>then </b>
<font color="Maroon">pk</font> is <a href="orders_3.html#V5">monotone</a>
 <b>by </b><i><a class="ref" href="waybel_1.html#D13">WAYBEL_1:def 13</a></i>;<br/><a NAME="E10:19_2_1"/><b>then </b><i><font color="Green">E84</font></i>: 
<font color="Maroon">pk</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font> is  <a href="waybel_0.html#V1">directed</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">Lk</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="yellow_2.html#T17">YELLOW_2:17</a></i>;<br/><a NAME="E11:19_2_1"/>
for <font color="Olive">Y</font> being non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">Lk</font></span>)</span>  holds not <font color="Maroon">pk</font> <a href="relset_1.html#K10">.:</a> <font color="Olive">Y</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><a NAME="E12:19_2_1"/><b>then </b><i><font color="Green">E86</font></i>: 
<font color="Maroon">pk</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i/>;<br/><a NAME="E13:19_2_1"/><i><font color="Green">E87</font></i>: 
 <a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">Lk</font> is <a href="waybel_0.html#V4">directed-sups-inheriting</a>
 <b>by </b><i>, <a class="ref" href="waybel_1.html#T55">WAYBEL_1:55</a></i>;<br/><a NAME="E14:19_2_1"/><b>then </b><i><font color="Green">E88</font></i>: 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">X'</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">X</font>
 <b>by </b><i>, , <a class="ref" href="waybel_0.html#T7">WAYBEL_0:7</a></i>;<br/><a NAME="E15:19_2_1"/>
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">Lk</font></span>)</span>
 ;<br/><a NAME="E16:19_2_1"/><b>then </b>
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Lk</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/><a NAME="E17:19_2_1"/><b>then </b>
<font color="Maroon">pk</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="extens_1.html#T1">EXTENS_1:1</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">pk</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">p</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font></span>)</span>
<b>by </b><i>, , <a class="txt" href="waybel15.html#E1:19"><i><font color="Green">E33</font></i></a>, <a class="txt" href="waybel15.html#E3:19"><i><font color="Green">E34</font></i></a>, <a class="ref" href="waybel_0.html#T7">WAYBEL_0:7</a></i>
<br/>.= 
<font color="Maroon">p</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="waybel15.html#T1" target="_self">Th1</a>, , <a class="ref" href="waybel_0.html#D31">WAYBEL_0:def 31</a></i>
<br/>.= 
<font color="Maroon">pk</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="waybel15.html#T2" target="_self">Th2</a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
;<br/><br/></div><b>end;</b></div><b>hence </b><a NAME="E6:19_2"/>
<font color="Maroon">pk</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>
<a NAME="E13:19"/><b>then </b>
<font color="Maroon">pk</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/>
<b>hence </b><a NAME="E14:19"/>
 <a href="yellow_2.html#K1">Image</a> <font color="Maroon">p</font> is  <a href="waybel_3.html#V3">continuous</a> <a href="waybel_0.html#NM8">LATTICE</a>
 <b>by </b><i>, , , , </i>;<br/>


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