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<b>let </b><font color="Maroon" title="c1">L</font>, <font color="Maroon" title="c2">M</font> be  <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a> <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>;<br/><b>let </b><font color="Maroon" title="c3">Lg</font>, <font color="Maroon" title="c4">Mg</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:26"/><i><font color="Green" title="E22">A1</font></i>: 
<font color="Maroon" title="c3">Lg</font> <a href="waybel22.html#R1" title="WAYBEL22:pred.1">is_FreeGen_set_of</a> <font color="Maroon" title="c1">L</font>
 <b>and </b><br/><a NAME="E2:26"/><i><font color="Green" title="E23">A2</font></i>: 
<font color="Maroon" title="c4">Mg</font> <a href="waybel22.html#R1" title="WAYBEL22:pred.1">is_FreeGen_set_of</a> <font color="Maroon" title="c2">M</font>
 <b>and </b><br/><a NAME="E3:26"/><i><font color="Green" title="E24">A3</font></i>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c3">Lg</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c4">Mg</font>
 ;<br/>

<a NAME="E4:26"/>
<font color="Maroon" title="c3">Lg</font>,<font color="Maroon" title="c4">Mg</font> <a href="tarski.html#R2" title="TARSKI:pred.2">are_equipotent</a> 
 
<b>by </b><i><a class="txt" href="waybel22.html#E3:26"><i><font color="Green" title="E24">A3</font></i></a>, <a class="ref" href="card_1.html#T21" title="CARD_1:th.21">CARD_1:21</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c5">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E6:26"/><i><font color="Green" title="E25">A4</font></i>: 
<font color="Maroon" title="c5">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>and </b><br/><a NAME="E7:26"/><i><font color="Green" title="E26">A5</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">Lg</font>
 <b>and </b><br/><a NAME="E8:26"/><i><font color="Green" title="E27">A6</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">Mg</font>
 <b>by </b><i><a class="ref" href="wellord2.html#D4" title="WELLORD2:def.4">WELLORD2:def 4</a></i>;<br/>
<b>set </b><font color="Maroon" title="c6">g</font> = <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> ;<br/>
<a NAME="E9:26"/><i><font color="Green" title="E28">A7</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">Mg</font>
 
<b>by </b><i><a class="txt" href="waybel22.html#E6:26"><i><font color="Green" title="E25">A4</font></i></a>, <a class="txt" href="waybel22.html#E8:26"><i><font color="Green" title="E27">A6</font></i></a>, <a class="ref" href="funct_1.html#T55" title="FUNCT_1:th.55">FUNCT_1:55</a></i>;<br/>
<a NAME="E10:26"/><i><font color="Green" title="E29">A8</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">Lg</font>
 
<b>by </b><i><a class="txt" href="waybel22.html#E6:26"><i><font color="Green" title="E25">A4</font></i></a>, <a class="txt" href="waybel22.html#E7:26"><i><font color="Green" title="E26">A5</font></i></a>, <a class="ref" href="funct_1.html#T55" title="FUNCT_1:th.55">FUNCT_1:55</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">Lg</font> = <font color="Maroon" title="c3">Lg</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="txt" href="waybel22.html#E1:26"><i><font color="Green" title="E22">A1</font></i></a>, <a class="ref" href="waybel22.html#T7" target="_self" title="WAYBEL22:th.7">Th7</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">Mg</font> = <font color="Maroon" title="c4">Mg</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">M</font> <b>by </b><i><a class="txt" href="waybel22.html#E2:26"><i><font color="Green" title="E23">A2</font></i></a>, <a class="ref" href="waybel22.html#T7" target="_self" title="WAYBEL22:th.7">Th7</a></i>;<br/>
<a NAME="E13:26"/>
<font color="Maroon" title="c8">Mg</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">M</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c9">f</font> = <font color="Maroon" title="c5">f</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c7">Lg</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">M</font> <b>by </b><i><a class="txt" href="waybel22.html#E7:26"><i><font color="Green" title="E26">A5</font></i></a>, <a class="txt" href="waybel22.html#E8:26"><i><font color="Green" title="E27">A6</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11" title="RELSET_1:th.11">RELSET_1:11</a></i>;<br/>
<a NAME="E15:26"/>
<font color="Maroon" title="c7">Lg</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c10">g</font> = <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c8">Mg</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="txt" href="waybel22.html#E9:26"><i><font color="Green" title="E28">A7</font></i></a>, <a class="txt" href="waybel22.html#E10:26"><i><font color="Green" title="E29">A8</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11" title="RELSET_1:th.11">RELSET_1:11</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c11">F</font> being    <a href="waybel16.html#M1" title="WAYBEL16:mode.1">CLHomomorphism</a> of <font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">M</font><b> such that </b><br/><a NAME="E18:26"/><i><font color="Green" title="E30">A9</font></i>: 
<font color="Maroon" title="c11">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c7">Lg</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f</font>
 <b>and </b><br/><a NAME="E19:26"/>
for <font color="Olive" title="b1">h'</font> being   <a href="waybel16.html#M1" title="WAYBEL16:mode.1">CLHomomorphism</a> of <font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">M</font>  st <font color="Olive" title="b1">h'</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c7">Lg</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f</font> holds <br/><font color="Olive" title="b1">h'</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c11">F</font>
 <b>by </b><i><a class="txt" href="waybel22.html#E1:26"><i><font color="Green" title="E22">A1</font></i></a>, <a class="ref" href="waybel22.html#D1" target="_self" title="WAYBEL22:def.1">Def1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">G</font> being    <a href="waybel16.html#M1" title="WAYBEL16:mode.1">CLHomomorphism</a> of <font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E21:26"/><i><font color="Green" title="E31">A10</font></i>: 
<font color="Maroon" title="c12">G</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c8">Mg</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">g</font>
 <b>and </b><br/><a NAME="E22:26"/>
for <font color="Olive" title="b1">h'</font> being   <a href="waybel16.html#M1" title="WAYBEL16:mode.1">CLHomomorphism</a> of <font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b1">h'</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c8">Mg</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">g</font> holds <br/><font color="Olive" title="b1">h'</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c12">G</font>
 <b>by </b><i><a class="txt" href="waybel22.html#E2:26"><i><font color="Green" title="E23">A2</font></i></a>, <a class="ref" href="waybel22.html#D1" target="_self" title="WAYBEL22:def.1">Def1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">GF</font> = <font color="Maroon" title="c12">G</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c11">F</font> as    <a href="waybel16.html#M1" title="WAYBEL16:mode.1">CLHomomorphism</a> of <font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c1">L</font> <b>by </b><i><a class="ref" href="waybel22.html#T2" target="_self" title="WAYBEL22:th.2">Th2</a></i>;<br/>
<font color="Maroon" title="c13">GF</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c7">Lg</font> = 
<font color="Maroon" title="c12">G</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c9">f</font>
<b>by </b><i><a class="txt" href="waybel22.html#E18:26"><i><font color="Green" title="E30">A9</font></i></a>, <a class="ref" href="relat_1.html#T112" title="RELAT_1:th.112">RELAT_1:112</a></i>
<br/>.= 
<font color="Maroon" title="c10">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c9">f</font>
<b>by </b><i><a class="txt" href="waybel22.html#E8:26"><i><font color="Green" title="E27">A6</font></i></a>, <a class="txt" href="waybel22.html#E21:26"><i><font color="Green" title="E31">A10</font></i></a>, <a class="ref" href="funct_4.html#T2" title="FUNCT_4:th.2">FUNCT_4:2</a></i>
<br/>.= 
 <a href="pralg_3.html#K1" title="PRALG_3:func.1">id</a> <font color="Maroon" title="c7">Lg</font>
<b>by </b><i><a class="txt" href="waybel22.html#E6:26"><i><font color="Green" title="E25">A4</font></i></a>, <a class="txt" href="waybel22.html#E7:26"><i><font color="Green" title="E26">A5</font></i></a>, <a class="ref" href="funct_1.html#T61" title="FUNCT_1:th.61">FUNCT_1:61</a></i>
;<br/>
<a NAME="E25:26"/><b>then </b><i><font color="Green" title="E32">A11</font></i>: 
<font color="Maroon" title="c13">GF</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="txt" href="waybel22.html#E1:26"><i><font color="Green" title="E22">A1</font></i></a>, <a class="ref" href="waybel22.html#T8" target="_self" title="WAYBEL22:th.8">Th8</a></i>;<br/>
<a NAME="E26:26"/><b>then </b><i><font color="Green" title="E33">A12</font></i>: 
<font color="Maroon" title="c11">F</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="funct_2.html#T29" title="FUNCT_2:th.29">FUNCT_2:29</a></i>;<br/>
<a NAME="E27:26"/>
<font color="Maroon" title="c11">F</font> is <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a>
 
<b>by </b><i><a class="ref" href="waybel16.html#D1" title="WAYBEL16:def.1">WAYBEL16:def 1</a></i>;<br/>
<a NAME="E28:26"/><b>then </b><i><font color="Green" title="E34">A13</font></i>: 
<font color="Maroon" title="c11">F</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a>
 
<b>by </b><i><a class="ref" href="waybel17.html#T3" title="WAYBEL17:th.3">WAYBEL17:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">FG</font> = <font color="Maroon" title="c11">F</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c12">G</font> as    <a href="waybel16.html#M1" title="WAYBEL16:mode.1">CLHomomorphism</a> of <font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c2">M</font> <b>by </b><i><a class="ref" href="waybel22.html#T2" target="_self" title="WAYBEL22:th.2">Th2</a></i>;<br/>
<font color="Maroon" title="c14">FG</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c8">Mg</font> = 
<font color="Maroon" title="c11">F</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c10">g</font>
<b>by </b><i><a class="txt" href="waybel22.html#E21:26"><i><font color="Green" title="E31">A10</font></i></a>, <a class="ref" href="relat_1.html#T112" title="RELAT_1:th.112">RELAT_1:112</a></i>
<br/>.= 
<font color="Maroon" title="c9">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c10">g</font>
<b>by </b><i><a class="txt" href="waybel22.html#E10:26"><i><font color="Green" title="E29">A8</font></i></a>, <a class="txt" href="waybel22.html#E18:26"><i><font color="Green" title="E30">A9</font></i></a>, <a class="ref" href="funct_4.html#T2" title="FUNCT_4:th.2">FUNCT_4:2</a></i>
<br/>.= 
 <a href="pralg_3.html#K1" title="PRALG_3:func.1">id</a> <font color="Maroon" title="c8">Mg</font>
<b>by </b><i><a class="txt" href="waybel22.html#E6:26"><i><font color="Green" title="E25">A4</font></i></a>, <a class="txt" href="waybel22.html#E8:26"><i><font color="Green" title="E27">A6</font></i></a>, <a class="ref" href="funct_1.html#T61" title="FUNCT_1:th.61">FUNCT_1:61</a></i>
;<br/>
<a NAME="E31:26"/><b>then </b>
<font color="Maroon" title="c14">FG</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">M</font>
 
<b>by </b><i><a class="txt" href="waybel22.html#E2:26"><i><font color="Green" title="E23">A2</font></i></a>, <a class="ref" href="waybel22.html#T8" target="_self" title="WAYBEL22:th.8">Th8</a></i>;<br/>
<a NAME="E32:26"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c11">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">M</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T29" title="FUNCT_2:th.29">FUNCT_2:29</a></i>;<br/>
<a NAME="E33:26"/><b>then </b><i><font color="Green" title="E35">A14</font></i>: 
<font color="Maroon" title="c12">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">F</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> 
 
<b>by </b><i><a class="txt" href="waybel22.html#E25:26"><i><font color="Green" title="E32">A11</font></i></a>, <a class="txt" href="waybel22.html#E26:26"><i><font color="Green" title="E33">A12</font></i></a>, <a class="ref" href="funct_2.html#T36" title="FUNCT_2:th.36">FUNCT_2:36</a></i>;<br/>
<a NAME="E34:26"/>
<font color="Maroon" title="c12">G</font> is <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a>
 
<b>by </b><i><a class="ref" href="waybel16.html#D1" title="WAYBEL16:def.1">WAYBEL16:def 1</a></i>;<br/>
<a NAME="E35:26"/><b>then </b>
<font color="Maroon" title="c12">G</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a>
 
<b>by </b><i><a class="ref" href="waybel17.html#T3" title="WAYBEL17:th.3">WAYBEL17:3</a></i>;<br/>
<a NAME="E36:26"/><b>then </b>
<font color="Maroon" title="c11">F</font> is <a href="waybel_0.html#V23" title="WAYBEL_0:attr.23">isomorphic</a>
 
<b>by </b><i><a class="txt" href="waybel22.html#E26:26"><i><font color="Green" title="E33">A12</font></i></a>, <a class="txt" href="waybel22.html#E28:26"><i><font color="Green" title="E34">A13</font></i></a>, <a class="txt" href="waybel22.html#E33:26"><i><font color="Green" title="E35">A14</font></i></a>, <a class="ref" href="waybel_0.html#D38" title="WAYBEL_0:def.38">WAYBEL_0:def 38</a></i>;<br/>
<b>hence </b><a NAME="E37:26"/>
<font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">M</font> <a href="waybel_1.html#R5" title="WAYBEL_1:pred.5">are_isomorphic</a> 
 <b>by </b><i><a class="ref" href="waybel_1.html#D8" title="WAYBEL_1:def.8">WAYBEL_1:def 8</a></i>;<br/>


</div>
