<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">L1</font> be  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#NM1">Poset</a>, <font color="Maroon">L2</font> be  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#NM1">Poset</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:12"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">L1</font>,<font color="Maroon">L2</font> <a href="waybel_1.html#R5">are_isomorphic</a> 
 <b>and </b><br/><a NAME="E2:12"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">L1</font> is <a href="waybel_3.html#V3">continuous</a>
 ;<br/>

<b>consider </b><font color="Maroon">f</font> being   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">L1</font>,<font color="Maroon">L2</font><b> such that </b><br/><a NAME="E4:12"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">f</font> is <a href="waybel_0.html#V23">isomorphic</a>
 <b>by </b><i>, <a class="ref" href="waybel_1.html#D8">WAYBEL_1:def 8</a></i>;<br/>
<a NAME="E5:12"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i>, <a class="ref" href="waybel_0.html#D38">WAYBEL_0:def 38</a></i>;<br/>
<b>reconsider </b><font color="Maroon">g</font> = <font color="Maroon">f</font> <a href="funct_1.html#K2">"</a>  as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">L2</font>,<font color="Maroon">L1</font> <b>by </b><i>, <a class="ref" href="waybel_0.html#T67">WAYBEL_0:67</a></i>;<br/>
<a NAME="E7:12"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">g</font> is <a href="waybel_0.html#V23">isomorphic</a>
 
<b>by </b><i>, <a class="ref" href="waybel_0.html#T68">WAYBEL_0:68</a></i>;<br/>
<a NAME="E8:12"/><i><font color="Green">E52</font></i>: 
( ( for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L1</font> holds <br/> ( not  <a href="waybel_3.html#K1">waybelow</a> <font color="Olive">x</font> is <a href="xboole_0.html#V1">empty</a> &amp;  <a href="waybel_3.html#K1">waybelow</a> <font color="Olive">x</font> is <a href="waybel_0.html#V1">directed</a> ) ) &amp; <font color="Maroon">L1</font> is <a href="waybel_0.html#V24">up-complete</a> &amp; <font color="Maroon">L1</font> is <a href="waybel_3.html#V2">satisfying_axiom_of_approximation</a> )
 
<b>by </b><i>, <a class="ref" href="waybel_3.html#D6">WAYBEL_3:def 6</a></i>;<br/>
<a NAME="E9:12"/><b>then </b><i><font color="Green">E53</font></i>: 
<font color="Maroon">L2</font> is <a href="waybel_0.html#V24">up-complete</a>
 
<b>by </b><i>, <a class="ref" href="waybel13.html#T30">WAYBEL13:30</a></i>;<br/>
<a NAME="E10:12"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">L1</font> is  non <a href="struct_0.html#V3">empty</a> <a href="waybel_0.html#V24">up-complete</a> <a href="orders_2.html#NM1">Poset</a> &amp; <font color="Maroon">L2</font> is  non <a href="struct_0.html#V3">empty</a> <a href="waybel_0.html#V24">up-complete</a> <a href="orders_2.html#NM1">Poset</a> )
 
<b>by </b><i>, , <a class="ref" href="waybel13.html#T30">WAYBEL13:30</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L2</font>;<br/><a NAME="E1:12_1"/>
<font color="Maroon">f</font> is <a href="waybel_0.html#V18">sups-preserving</a>
 <b>by </b><i>, <a class="ref" href="waybel13.html#T20">WAYBEL13:20</a></i>;<br/><a NAME="E2:12_1"/><b>then </b><i><font color="Green">E56</font></i>: 
<font color="Maroon">f</font> <a href="waybel_0.html#R4">preserves_sup_of</a>  <a href="waybel_3.html#K1">waybelow</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span>
 <b>by </b><i><a class="ref" href="waybel_0.html#D33">WAYBEL_0:def 33</a></i>;<br/><a NAME="E3:12_1"/>
 <a href="waybel_3.html#K1">waybelow</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span> is  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">L1</font>
 <b>by </b><i>, <a class="ref" href="waybel_3.html#D6">WAYBEL_3:def 6</a></i>;<br/><a NAME="E4:12_1"/><b>then </b><i><font color="Green">E57</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a>  <a href="waybel_3.html#K1">waybelow</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span>,<font color="Maroon">L1</font>
 <b>by </b><i>, <a class="ref" href="waybel_0.html#T75">WAYBEL_0:75</a></i>;<br/><a NAME="E5:12_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L2</font>
 ;<br/><a NAME="E6:12_1"/><b>then </b><i><font color="Green">E69</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="waybel_0.html#T66">WAYBEL_0:66</a></i>;<br/><b>hence </b><font color="Maroon">y</font> = 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_1.html#T57">FUNCT_1:57</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p2">(<span class="default"><a href="waybel_3.html#K1">waybelow</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="waybel_3.html#D5">WAYBEL_3:def 5</a></i>
<br/>.= 
 <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> <span class="p2">(<span class="default"><a href="waybel_3.html#K1">waybelow</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="waybel_0.html#D31">WAYBEL_0:def 31</a></i>
<br/>.= 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="waybel_3.html#K1">waybelow</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T57">FUNCT_1:57</a></i>
;<br/><br/></div><b>end;</b></div>
<a NAME="E12:12"/><b>then </b><i><font color="Green">E70</font></i>: 
<font color="Maroon">L2</font> is <a href="waybel_3.html#V2">satisfying_axiom_of_approximation</a>
 
<b>by </b><i><a class="ref" href="waybel_3.html#D5">WAYBEL_3:def 5</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L2</font>;<br/><a NAME="E1:12_2"/>
for <font color="Olive">Y</font> being <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">y</font></span>)</span>  ex <font color="Olive">z</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L2</font> st <br/>( <font color="Olive">z</font> <a href="hidden.html#R2">in</a>  <a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">y</font> &amp; <font color="Olive">z</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Olive">Y</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">Y</font> be  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">y</font></span>)</span>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">u</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:12_2_1_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">Y</font>
 ;<br/><b>then consider </b><font color="Maroon">v</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:12_2_1_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font>
 <b>and </b><br/><a NAME="E4:12_2_1_1"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>and </b><br/><a NAME="E5:12_2_1_1"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/><a NAME="E6:12_2_1_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">y</font>
 <b>by </b><i/>;<br/><a NAME="E7:12_2_1_1"/><b>then </b>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <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">L2</font> : <font color="Olive">k</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">y</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="waybel_3.html#D3">WAYBEL_3:def 3</a></i>;<br/><b>then consider </b><font color="Maroon">v1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L2</font><b> such that </b><br/><a NAME="E9:12_2_1_1"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font>
 <b>and </b><br/><a NAME="E10:12_2_1_1"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">v1</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">y</font>
 ;<br/><a NAME="E11:12_2_1_1"/>
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v1</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>
 <b>by </b><i>, , , <a class="ref" href="waybel13.html#T27">WAYBEL13:27</a></i>;<br/><b>hence </b><a NAME="E12:12_2_1_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="waybel_3.html#K1">waybelow</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="waybel_3.html#T7">WAYBEL_3:7</a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">X</font> = <font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">Y</font> as  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="waybel_3.html#K1">waybelow</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E3:12_2_1"/>
( not  <a href="waybel_3.html#K1">waybelow</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span> is <a href="xboole_0.html#V1">empty</a> &amp;  <a href="waybel_3.html#K1">waybelow</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span> is <a href="waybel_0.html#V1">directed</a> )
 
<b>by </b><i>, <a class="ref" href="waybel_3.html#D6">WAYBEL_3:def 6</a></i>;<br/>
<b>then consider </b><font color="Maroon">x</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L1</font><b> such that </b><br/><a NAME="E5:12_2_1"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="waybel_3.html#K1">waybelow</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span>
 <b>and </b><br/><a NAME="E6:12_2_1"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">x</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="waybel_0.html#T1">WAYBEL_0:1</a></i>;<br/>
<b>take </b><font color="Maroon">z</font> = <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>;<br/>

<a NAME="E7:12_2_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L2</font>
 
;<br/>
<a NAME="E8:12_2_1"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="waybel_0.html#T66">WAYBEL_0:66</a></i>;<br/>
<a NAME="E9:12_2_1"/><b>then </b><i><font color="Green">E90</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T57">FUNCT_1:57</a></i>;<br/>
<a NAME="E10:12_2_1"/>
<font color="Maroon">x</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>
 
<b>by </b><i>, <a class="ref" href="waybel_3.html#T7">WAYBEL_3:7</a></i>;<br/>
<a NAME="E11:12_2_1"/><b>then </b>
<font color="Maroon">z</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">y</font>
 
<b>by </b><i>, , , <a class="ref" href="waybel13.html#T27">WAYBEL13:27</a></i>;<br/>
<b>hence </b><a NAME="E12:12_2_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="waybel_3.html#T7">WAYBEL_3:7</a></i>;<br/>

<a NAME="E13:12_2_1"/>
<font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L2</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E14:12_2_1"/><b>then </b><i><font color="Green">E91</font></i>: 
<font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="waybel_0.html#T66">WAYBEL_0:66</a></i>;<br/>
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font> = 
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">Y</font></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_1.html#T155">FUNCT_1:155</a></i>
<br/>.= 
<font color="Maroon">Y</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T147">FUNCT_1:147</a></i>
;<br/>
<b>hence </b><a NAME="E16:12_2_1"/>
<font color="Maroon">z</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">Y</font>
 <b>by </b><i>, , <a class="ref" href="waybel13.html#T19">WAYBEL13:19</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E2:12_2"/>
( not  <a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">y</font> is <a href="xboole_0.html#V1">empty</a> &amp;  <a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">y</font> is <a href="waybel_0.html#V1">directed</a> )
 <b>by </b><i><a class="ref" href="waybel_0.html#T1">WAYBEL_0:1</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E14:12"/>
<font color="Maroon">L2</font> is <a href="waybel_3.html#V3">continuous</a>
 <b>by </b><i>, , <a class="ref" href="waybel_3.html#D6">WAYBEL_3:def 6</a></i>;<br/>


</div>
