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

<b>assume </b><b>that </b><br/><a NAME="E1:19"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">L</font>,<font color="Maroon">M</font> <a href="waybel_1.html#R5">are_isomorphic</a> 
 <b>and </b><br/><a NAME="E2:19"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">L</font> is <a href="lattice3.html#V3">complete</a>
 ;<br/>

<a NAME="E3:19"/>
<font color="Maroon">M</font>,<font color="Maroon">L</font> <a href="waybel_1.html#R5">are_isomorphic</a> 
 
<b>by </b><i>, <a class="ref" href="waybel_1.html#T7">WAYBEL_1:7</a></i>;<br/>
<b>then consider </b><font color="Maroon">f</font> being   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">M</font>,<font color="Maroon">L</font><b> such that </b><br/><a NAME="E5:19"/><i><font color="Green">E37</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="E6:19"/><i><font color="Green">E39</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#T66">WAYBEL_0:66</a></i>;<br/>
<a NAME="E7:19"/><i><font color="Green">E40</font></i>: 
 <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">L</font>
 
<b>by </b><i>, <a class="ref" href="waybel_0.html#T66">WAYBEL_0:66</a></i>;<br/>
<a NAME="E8:19"/><i><font color="Green">E41</font></i>: 
 <a href="relset_1.html#K4">dom</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#D1">FUNCT_2:def 1</a></i>;<br/>
<b>let </b><font color="Maroon">X</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">M</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice5.html#D2">LATTICE5:def 2</a><br/>

<b>reconsider </b><font color="Maroon">fX</font> = <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> ;<br/>
<b>consider </b><font color="Maroon">fa</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E11:19"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">fa</font> <a href="lattice3.html#R1">is_&lt;=_than</a> <font color="Maroon">fX</font>
 <b>and </b><br/><a NAME="E12:19"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">fb</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  st <font color="Olive">fb</font> <a href="lattice3.html#R1">is_&lt;=_than</a> <font color="Maroon">fX</font> holds <br/><font color="Olive">fb</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">fa</font>
 <b>by </b><i>, <a class="ref" href="lattice5.html#D2">LATTICE5:def 2</a></i>;<br/>
<b>set </b><font color="Maroon">a</font> = <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">fa</font>;<br/>
<a NAME="E13:19"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">fa</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#T54">FUNCT_1:54</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">a</font> = <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">fa</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font> ;<br/>
<a NAME="E15:19"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">fa</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#T57">FUNCT_1:57</a></i>;<br/>
<b>take </b>
<font color="Maroon">a</font>
;<br/>

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D8">LATTICE3:def 8</a>
<div class="add"><b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font>;<br/><b>assume </b><a NAME="E1:19_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><b>reconsider </b><font color="Maroon">fb</font> = <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ;<br/><a NAME="E3:19_1"/>
<font color="Maroon">fb</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fX</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/><a NAME="E4:19_1"/><b>then </b>
<font color="Maroon">fa</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">fb</font>
 <b>by </b><i>, <a class="ref" href="lattice3.html#D8">LATTICE3:def 8</a></i>;<br/><b>hence </b><a NAME="E5:19_1"/>
<font color="Maroon">a</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">b</font>
 <b>by </b><i>, <a class="ref" href="waybel20.html#T1" target="_self">Th1</a>, <a class="ref" href="waybel_0.html#T66">WAYBEL_0:66</a></i>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font>;<br/>

<b>assume </b><a NAME="E17:19"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">b</font> <a href="lattice3.html#R1">is_&lt;=_than</a> <font color="Maroon">X</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">fb</font> = <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ;<br/>
<a NAME="E19:19"/>
<font color="Maroon">fb</font> <a href="lattice3.html#R1">is_&lt;=_than</a> <font color="Maroon">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">fc</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D8">LATTICE3:def 8</a><br/>

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

<b>then consider </b><font color="Maroon">c</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:19_2"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">fc</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c</font> = <font color="Maroon">c</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font> <b>by </b><i/>;<br/>
<a NAME="E5:19_2"/>
<font color="Maroon">b</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c</font>
 
<b>by </b><i>, , <a class="ref" href="lattice3.html#D8">LATTICE3:def 8</a></i>;<br/>
<b>hence </b><a NAME="E6:19_2"/>
<font color="Maroon">fb</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">fc</font>
 <b>by </b><i>, , <a class="ref" href="waybel_0.html#T66">WAYBEL_0:66</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E20:19"/><b>then </b>
<font color="Maroon">fb</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">fa</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E21:19"/>
<font color="Maroon">b</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="ref" href="waybel20.html#T1" target="_self">Th1</a>, <a class="ref" href="waybel_0.html#T66">WAYBEL_0:66</a></i>;<br/>


</div>
