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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>, <font color="Maroon">M</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">Lg</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">Mg</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:26"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">Lg</font> <a href="waybel22.html#R1" target="_self">is_FreeGen_set_of</a> <font color="Maroon">L</font>
 <b>and </b><br/><a NAME="E2:26"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">Mg</font> <a href="waybel22.html#R1" target="_self">is_FreeGen_set_of</a> <font color="Maroon">M</font>
 <b>and </b><br/><a NAME="E3:26"/><i><font color="Green">E37</font></i>: 
 <a href="card_1.html#K1">Card</a> <font color="Maroon">Lg</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">Mg</font>
 ;<br/>

<a NAME="E4:26"/>
<font color="Maroon">Lg</font>,<font color="Maroon">Mg</font> <a href="tarski.html#R2">are_equipotent</a> 
 
<b>by </b><i>, <a class="ref" href="card_1.html#T21">CARD_1:21</a></i>;<br/>
<b>then consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E6:26"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>and </b><br/><a NAME="E7:26"/><i><font color="Green">E59</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Lg</font>
 <b>and </b><br/><a NAME="E8:26"/><i><font color="Green">E60</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Mg</font>
 <b>by </b><i><a class="ref" href="wellord2.html#D4">WELLORD2:def 4</a></i>;<br/>
<b>set </b><font color="Maroon">g</font> = <font color="Maroon">f</font> <a href="funct_1.html#K2">"</a> ;<br/>
<a NAME="E9:26"/><i><font color="Green">E61</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Mg</font>
 
<b>by </b><i>, <a class="txt" href="waybel22.html#E1"><i><font color="Green">Lemma27</font></i></a>, <a class="ref" href="funct_1.html#T55">FUNCT_1:55</a></i>;<br/>
<a NAME="E10:26"/><i><font color="Green">E62</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Lg</font>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#T55">FUNCT_1:55</a></i>;<br/>
<b>reconsider </b><font color="Maroon">Lg</font> = <font color="Maroon">Lg</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> <b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">Mg</font> = <font color="Maroon">Mg</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">M</font> <b>by </b><i>, </i>;<br/>
<a NAME="E13:26"/>
<font color="Maroon">Mg</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">Lg</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font> <b>by </b><i>, <a class="txt" href="waybel22.html#E1"><i><font color="Green">Lemma27</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<a NAME="E15:26"/>
<font color="Maroon">Lg</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">g</font> = <font color="Maroon">f</font> <a href="funct_1.html#K2">"</a>  as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">Mg</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> <b>by </b><i>, , <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<b>consider </b><font color="Maroon">F</font> being    <a href="waybel16.html#M1">CLHomomorphism</a> of <font color="Maroon">L</font>,<font color="Maroon">M</font><b> such that </b><br/><a NAME="E18:26"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">Lg</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E19:26"/>
for <font color="Olive">h'</font> being   <a href="waybel16.html#M1">CLHomomorphism</a> of <font color="Maroon">L</font>,<font color="Maroon">M</font>  st <font color="Olive">h'</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">Lg</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> holds <br/><font color="Olive">h'</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">F</font>
 <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">G</font> being    <a href="waybel16.html#M1">CLHomomorphism</a> of <font color="Maroon">M</font>,<font color="Maroon">L</font><b> such that </b><br/><a NAME="E21:26"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">G</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">Mg</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font>
 <b>and </b><br/><a NAME="E22:26"/>
for <font color="Olive">h'</font> being   <a href="waybel16.html#M1">CLHomomorphism</a> of <font color="Maroon">M</font>,<font color="Maroon">L</font>  st <font color="Olive">h'</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">Mg</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> holds <br/><font color="Olive">h'</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">G</font>
 <b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">GF</font> = <font color="Maroon">G</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font> as    <a href="waybel16.html#M1">CLHomomorphism</a> of <font color="Maroon">L</font>,<font color="Maroon">L</font> <b>by </b><i/>;<br/>
<font color="Maroon">GF</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">Lg</font> = 
<font color="Maroon">G</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font>
<b>by </b><i>, <a class="ref" href="relat_1.html#T112">RELAT_1:112</a></i>
<br/>.= 
<font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font>
<b>by </b><i><a class="txt" href="waybel22.html#E1"><i><font color="Green">Lemma27</font></i></a>, <a class="txt" href="waybel22.html#E1:26"><i><font color="Green">E31</font></i></a>, <a class="ref" href="funct_4.html#T2">FUNCT_4:2</a></i>
<br/>.= 
 <a href="pralg_3.html#K1">id</a> <font color="Maroon">Lg</font>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T61">FUNCT_1:61</a></i>
;<br/>
<a NAME="E25:26"/><b>then </b>
<font color="Maroon">GF</font> <a href="funct_2.html#R2">=</a>  <a href="grcat_1.html#K7">id</a> <font color="Maroon">L</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E26:26"/><b>then </b><i><font color="Green">E67</font></i>: 
<font color="Maroon">G</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="pralg_3.html#K1">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 
;<br/>
<a NAME="E27:26"/><b>then </b><i><font color="Green">E68</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="funct_2.html#T29">FUNCT_2:29</a></i>;<br/>
<a NAME="E28:26"/>
<font color="Maroon">F</font> is <a href="waybel_0.html#V22">directed-sups-preserving</a>
 
<b>by </b><i><a class="ref" href="waybel16.html#D1">WAYBEL16:def 1</a></i>;<br/>
<a NAME="E29:26"/><b>then </b><i><font color="Green">E69</font></i>: 
<font color="Maroon">F</font> is <a href="orders_3.html#V5">monotone</a>
 
<b>by </b><i><a class="ref" href="waybel17.html#T3">WAYBEL17:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">FG</font> = <font color="Maroon">F</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">G</font> as    <a href="waybel16.html#M1">CLHomomorphism</a> of <font color="Maroon">M</font>,<font color="Maroon">M</font> <b>by </b><i/>;<br/>
<font color="Maroon">FG</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">Mg</font> = 
<font color="Maroon">F</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">g</font>
<b>by </b><i><a class="txt" href="waybel22.html#E1:26"><i><font color="Green">E31</font></i></a>, <a class="ref" href="relat_1.html#T112">RELAT_1:112</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">g</font>
<b>by </b><i>, , <a class="ref" href="funct_4.html#T2">FUNCT_4:2</a></i>
<br/>.= 
 <a href="pralg_3.html#K1">id</a> <font color="Maroon">Mg</font>
<b>by </b><i>, <a class="txt" href="waybel22.html#E1"><i><font color="Green">Lemma27</font></i></a>, <a class="ref" href="funct_1.html#T61">FUNCT_1:61</a></i>
;<br/>
<a NAME="E32:26"/><b>then </b>
<font color="Maroon">FG</font> <a href="funct_2.html#R2">=</a>  <a href="grcat_1.html#K7">id</a> <font color="Maroon">M</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E33:26"/><b>then </b>
<font color="Maroon">F</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="pralg_3.html#K1">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font>
 
;<br/>
<a NAME="E34:26"/><b>then </b>
 <a href="relset_1.html#K5">rng</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">M</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T29">FUNCT_2:29</a></i>;<br/>
<a NAME="E35:26"/><b>then </b><i><font color="Green">E70</font></i>: 
<font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K2">"</a> 
 
<b>by </b><i>, , <a class="ref" href="funct_2.html#T36">FUNCT_2:36</a></i>;<br/>
<a NAME="E36:26"/>
<font color="Maroon">G</font> is <a href="waybel_0.html#V22">directed-sups-preserving</a>
 
<b>by </b><i><a class="ref" href="waybel16.html#D1">WAYBEL16:def 1</a></i>;<br/>
<a NAME="E37:26"/><b>then </b>
<font color="Maroon">G</font> is <a href="orders_3.html#V5">monotone</a>
 
<b>by </b><i><a class="ref" href="waybel17.html#T3">WAYBEL17:3</a></i>;<br/>
<a NAME="E38:26"/><b>then </b>
<font color="Maroon">F</font> is <a href="waybel_0.html#V23">isomorphic</a>
 
<b>by </b><i>, , <a class="txt" href="waybel22.html#E2:26"><i><font color="Green">E35</font></i></a>, <a class="ref" href="waybel_0.html#D38">WAYBEL_0:def 38</a></i>;<br/>
<b>hence </b><a NAME="E39:26"/>
<font color="Maroon">L</font>,<font color="Maroon">M</font> <a href="waybel_1.html#R5">are_isomorphic</a> 
 <b>by </b><i><a class="ref" href="waybel_1.html#D8">WAYBEL_1:def 8</a></i>;<br/>


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