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<b>let </b><font color="Maroon" title="c1">L</font>, <font color="Maroon" title="c2">M</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:19"/><i><font color="Green" title="E17">A1</font></i>: 
<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>and </b><br/><a NAME="E2:19"/><i><font color="Green" title="E18">A2</font></i>: 
<font color="Maroon" title="c1">L</font> is <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a>
 ;<br/>

<a NAME="E3:19"/>
<font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c1">L</font> <a href="waybel_1.html#R5" title="WAYBEL_1:pred.5">are_isomorphic</a> 
 
<b>by </b><i><a class="txt" href="waybel20.html#E1:19"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="waybel_1.html#T7" title="WAYBEL_1:th.7">WAYBEL_1:7</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c3">f</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E5:19"/><i><font color="Green" title="E19">A3</font></i>: 
<font color="Maroon" title="c3">f</font> is <a href="waybel_0.html#V23" title="WAYBEL_0:attr.23">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/>
<a NAME="E6:19"/><i><font color="Green" title="E20">A4</font></i>: 
<font color="Maroon" title="c3">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="waybel20.html#E5:19"><i><font color="Green" title="E19">A3</font></i></a>, <a class="ref" href="waybel_0.html#T66" title="WAYBEL_0:th.66">WAYBEL_0:66</a></i>;<br/>
<a NAME="E7:19"/><i><font color="Green" title="E21">A5</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">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="c1">L</font>
 
<b>by </b><i><a class="txt" href="waybel20.html#E5:19"><i><font color="Green" title="E19">A3</font></i></a>, <a class="ref" href="waybel_0.html#T66" title="WAYBEL_0:th.66">WAYBEL_0:66</a></i>;<br/>
<a NAME="E8:19"/><i><font color="Green" title="E22">A6</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">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#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>let </b><font color="Maroon" title="c4">X</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">M</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice5.html#D2" title="LATTICE5:def.2">LATTICE5:def 2</a><br/>

<b>reconsider </b><font color="Maroon" title="c5">fX</font> = <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c4">X</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> ;<br/>
<b>consider </b><font color="Maroon" title="c6">fa</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E11:19"/><i><font color="Green" title="E23">A7</font></i>: 
<font color="Maroon" title="c6">fa</font> <a href="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a> <font color="Maroon" title="c5">fX</font>
 <b>and </b><br/><a NAME="E12:19"/><i><font color="Green" title="E24">A8</font></i>: 
for <font color="Olive" title="b1">fb</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b1">fb</font> <a href="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a> <font color="Maroon" title="c5">fX</font> holds <br/><font color="Olive" title="b1">fb</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c6">fa</font>
 <b>by </b><i><a class="txt" href="waybel20.html#E2:19"><i><font color="Green" title="E18">A2</font></i></a>, <a class="ref" href="lattice5.html#D2" title="LATTICE5:def.2">LATTICE5:def 2</a></i>;<br/>
<b>set </b><font color="Maroon" title="c7">a</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">fa</font>;<br/>
<a NAME="E13:19"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">fa</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font>
 
<b>by </b><i><a class="txt" href="waybel20.html#E6:19"><i><font color="Green" title="E20">A4</font></i></a>, <a class="txt" href="waybel20.html#E7:19"><i><font color="Green" title="E21">A5</font></i></a>, <a class="ref" href="funct_1.html#T54" title="FUNCT_1:th.54">FUNCT_1:54</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c8">a</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">fa</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">M</font> ;<br/>
<a NAME="E15:19"/><i><font color="Green" title="E25">A9</font></i>: 
<font color="Maroon" title="c6">fa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">a</font>
 
<b>by </b><i><a class="txt" href="waybel20.html#E6:19"><i><font color="Green" title="E20">A4</font></i></a>, <a class="txt" href="waybel20.html#E7:19"><i><font color="Green" title="E21">A5</font></i></a>, <a class="ref" href="funct_1.html#T57" title="FUNCT_1:th.57">FUNCT_1:57</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c8">a</font>
;<br/>

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D8" title="LATTICE3:def.8">LATTICE3:def 8</a>
<div class="add"><b>let </b><font color="Maroon" title="c9">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">M</font>;<br/><b>assume </b><a NAME="E1:19_1"/><i><font color="Green" title="E26">A10</font></i>: 
<font color="Maroon" title="c9">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 ;<br/><b>reconsider </b><font color="Maroon" title="c10">fb</font> = <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">b</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> ;<br/><a NAME="E3:19_1"/>
<font color="Maroon" title="c10">fb</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">fX</font>
 <b>by </b><i><a class="txt" href="waybel20.html#E8:19"><i><font color="Green" title="E22">A6</font></i></a>, <a class="txt" href="waybel20.html#E1:19_1"><i><font color="Green" title="E26">A10</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/><a NAME="E4:19_1"/><b>then </b>
<font color="Maroon" title="c6">fa</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c10">fb</font>
 <b>by </b><i><a class="txt" href="waybel20.html#E11:19"><i><font color="Green" title="E23">A7</font></i></a>, <a class="ref" href="lattice3.html#D8" title="LATTICE3:def.8">LATTICE3:def 8</a></i>;<br/><b>hence </b><a NAME="E5:19_1"/>
<font color="Maroon" title="c8">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c9">b</font>
 <b>by </b><i><a class="txt" href="waybel20.html#E5:19"><i><font color="Green" title="E19">A3</font></i></a>, <a class="txt" href="waybel20.html#E15:19"><i><font color="Green" title="E25">A9</font></i></a>, <a class="ref" href="waybel_0.html#T66" title="WAYBEL_0:th.66">WAYBEL_0:66</a></i>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon" title="c9">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">M</font>;<br/>

<b>assume </b><a NAME="E17:19"/><i><font color="Green" title="E26">A11</font></i>: 
<font color="Maroon" title="c9">b</font> <a href="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a> <font color="Maroon" title="c4">X</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c10">fb</font> = <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">b</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> ;<br/>
<a NAME="E19:19"/>
<font color="Maroon" title="c10">fb</font> <a href="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a> <font color="Maroon" title="c5">fX</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">fc</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D8" title="LATTICE3:def.8">LATTICE3:def 8</a><br/>

<b>assume </b><a NAME="E1:19_2"/>
<font color="Maroon" title="c11">fc</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">fX</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c12">c</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:19_2"/><i><font color="Green" title="E27">A12</font></i>: 
( <font color="Maroon" title="c12">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> &amp; <font color="Maroon" title="c12">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> &amp; <font color="Maroon" title="c11">fc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">c</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">c</font> = <font color="Maroon" title="c12">c</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">M</font> <b>by </b><i><a class="txt" href="waybel20.html#E3:19_2"><i><font color="Green" title="E27">A12</font></i></a></i>;<br/>
<a NAME="E5:19_2"/>
<font color="Maroon" title="c9">b</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c13">c</font>
 
<b>by </b><i><a class="txt" href="waybel20.html#E17:19"><i><font color="Green" title="E26">A11</font></i></a>, <a class="txt" href="waybel20.html#E3:19_2"><i><font color="Green" title="E27">A12</font></i></a>, <a class="ref" href="lattice3.html#D8" title="LATTICE3:def.8">LATTICE3:def 8</a></i>;<br/>
<b>hence </b><a NAME="E6:19_2"/>
<font color="Maroon" title="c10">fb</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c11">fc</font>
 <b>by </b><i><a class="txt" href="waybel20.html#E5:19"><i><font color="Green" title="E19">A3</font></i></a>, <a class="txt" href="waybel20.html#E3:19_2"><i><font color="Green" title="E27">A12</font></i></a>, <a class="ref" href="waybel_0.html#T66" title="WAYBEL_0:th.66">WAYBEL_0:66</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E20:19"/><b>then </b>
<font color="Maroon" title="c10">fb</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c6">fa</font>
 
<b>by </b><i><a class="txt" href="waybel20.html#E12:19"><i><font color="Green" title="E24">A8</font></i></a></i>;<br/>
<b>hence </b><a NAME="E21:19"/>
<font color="Maroon" title="c9">b</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c8">a</font>
 <b>by </b><i><a class="txt" href="waybel20.html#E5:19"><i><font color="Green" title="E19">A3</font></i></a>, <a class="txt" href="waybel20.html#E15:19"><i><font color="Green" title="E25">A9</font></i></a>, <a class="ref" href="waybel_0.html#T66" title="WAYBEL_0:th.66">WAYBEL_0:66</a></i>;<br/>


</div>
