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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#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="E1:37"/><i><font color="Green">E30</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="E2:37"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">L1</font> is <a href="waybel_0.html#V24">up-complete</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="E4:37"/><i><font color="Green">E35</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="E5:37"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="waybel13.html#E2:37"><i><font color="Green">E34</font></i></a>, <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="txt" href="waybel13.html#E2:37"><i><font color="Green">E34</font></i></a>, <a class="ref" href="waybel_0.html#T67">WAYBEL_0:67</a></i>;<br/>
<a NAME="E7:37"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">g</font> is <a href="waybel_0.html#V23">isomorphic</a>
 
<b>by </b><i><a class="txt" href="waybel13.html#E2:37"><i><font color="Green">E34</font></i></a>, <a class="ref" href="waybel_0.html#T68">WAYBEL_0:68</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">Y</font> be  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">L2</font>;<br/><a NAME="E1:37_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>
 ;<br/><a NAME="E2:37_1"/><b>then </b><i><font color="Green">E50</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="txt" href="waybel13.html#E2:37"><i><font color="Green">E34</font></i></a>, <a class="ref" href="waybel_0.html#T66">WAYBEL_0:66</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">X1</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"><font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">Y</font></span>)</span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">v</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:37_1_1_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X1</font>
 ;<br/><b>then consider </b><font color="Maroon">u</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:37_1_1_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E4:37_1_1_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font>
 <b>and </b><br/><a NAME="E5:37_1_1_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/><a NAME="E6:37_1_1_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">Y</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="waybel13.html#T3" target="_self">Th3</a>, , <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/><a NAME="E7:37_1_1_1"/><b>then </b>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <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="txt" href="waybel13.html#E4:37"><i><font color="Green">E35</font></i></a>, <a class="ref" href="funct_1.html#T155">FUNCT_1:155</a></i>;<br/><b>hence </b><a NAME="E8:37_1_1_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="waybel13.html#T2" target="_self">Th2</a>, <a class="ref" href="funct_1.html#T147">FUNCT_1:147</a></i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">Y1</font> = <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X1</font> as  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Y</font> <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><b>consider </b><font color="Maroon">y1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L2</font><b> such that </b><br/><a NAME="E4:37_1_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>and </b><br/><a NAME="E5:37_1_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">y1</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">Y1</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">gy1</font> = <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y1</font>;<br/><a NAME="E6:37_1_1"/>
( not the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L1</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">y1</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="E7:37_1_1"/><b>then </b>
<font color="Maroon">y1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><b>hence </b><a NAME="E8:37_1_1"/>
<font color="Maroon">gy1</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>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/><a NAME="E9:37_1_1"/>
( <font color="Maroon">X1</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L1</font> &amp; 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> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E10:37_1_1"/><b>then </b>
<font color="Maroon">X1</font> <a href="tarski.html#R1">c=</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="E11:37_1_1"/><b>then </b><i><font color="Green">E61</font></i>: 
<font color="Maroon">X1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X1</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T146">FUNCT_1:146</a></i>;<br/><a NAME="E12:37_1_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X1</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">X1</font>
 <b>by </b><i><a class="txt" href="waybel13.html#E4:37"><i><font color="Green">E35</font></i></a>, <a class="ref" href="funct_1.html#T152">FUNCT_1:152</a></i>;<br/><font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">Y1</font> = 
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X1</font></span>)</span>
<b>by </b><i><a class="txt" href="waybel13.html#E4:37"><i><font color="Green">E35</font></i></a>, <a class="ref" href="funct_1.html#T155">FUNCT_1:155</a></i>
<br/>.= 
<font color="Maroon">X1</font>
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>
;<br/><b>hence </b><a NAME="E14:37_1_1"/>
<font color="Maroon">gy1</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">X1</font>
 <b>by </b><i><a class="txt" href="waybel13.html#E5:37"><i><font color="Green">E36</font></i></a>, <a class="ref" href="waybel13.html#T3" target="_self">Th3</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  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_0.html#T1">WAYBEL_0:1</a></i>;<br/><a NAME="E5:37_1"/>
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">X</font>,<font color="Maroon">L1</font>
 <b>by </b><i>, <a class="ref" href="waybel_0.html#T75">WAYBEL_0:75</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="E7:37_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">X</font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E8:37_1"/><i><font color="Green">E65</font></i>: 
for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L1</font>  st <font color="Maroon">X</font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Olive">b</font> holds <br/><font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">b</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T15">YELLOW_0:15</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="txt" href="waybel13.html#E4:37"><i><font color="Green">E35</font></i></a>, <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="waybel13.html#T2" target="_self">Th2</a>, <a class="ref" href="funct_1.html#T147">FUNCT_1:147</a></i>
;<br/><a NAME="E10:37_1"/><b>then </b><i><font color="Green">E68</font></i>: 
<font color="Maroon">Y</font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="txt" href="waybel13.html#E2:37"><i><font color="Green">E34</font></i></a>, , </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_3"><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="E1:37_1_3"/>
( not the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L1</font> is <a href="xboole_0.html#V1">empty</a> &amp; <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="E2:37_1_3"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E3:37_1_3"/><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="txt" href="waybel13.html#E4:37"><i><font color="Green">E35</font></i></a>, <a class="ref" href="funct_1.html#T55">FUNCT_1:55</a></i>;<br/><b>assume </b><a NAME="E4:37_1_3"/>
<font color="Maroon">Y</font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Maroon">y</font>
 ;<br/><a NAME="E5:37_1_3"/><b>then </b>
<font color="Maroon">X</font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="txt" href="waybel13.html#E5:37"><i><font color="Green">E36</font></i></a>, </i>;<br/><a NAME="E6:37_1_3"/><b>then </b>
<font color="Maroon">x</font> <a href="orders_2.html#R3">&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#T3" target="_self">Th3</a></i>;<br/><a NAME="E7:37_1_3"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <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="txt" href="waybel13.html#E2:37"><i><font color="Green">E34</font></i></a>, <a class="ref" href="waybel_0.html#T66">WAYBEL_0:66</a></i>;<br/><b>hence </b><a NAME="E8:37_1_3"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="txt" href="waybel13.html#E4:37"><i><font color="Green">E35</font></i></a>, , <a class="ref" href="funct_1.html#T57">FUNCT_1:57</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E12:37_1"/>
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">Y</font>,<font color="Maroon">L2</font>
 <b>by </b><i>, <a class="ref" href="yellow_0.html#T15">YELLOW_0:15</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E9:37"/>
<font color="Maroon">L2</font> is <a href="waybel_0.html#V24">up-complete</a>
 <b>by </b><i><a class="ref" href="waybel_0.html#T75">WAYBEL_0:75</a></i>;<br/>


</div>
