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<div class="add">

<b>set </b><font color="Maroon">f</font> =  <a href="yellow_7.html#K1" target="_self">ComplMap</a> <font color="Maroon">L</font>;<br/>
<a NAME="E1:51_1_1"/><i><font color="Green">E32</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="yellow_7.html#K1" target="_self">ComplMap</a> <font color="Maroon">L</font></span>)</span> <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="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E2:51_1_1"/><i><font color="Green">E33</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="yellow_7.html#K1" target="_self">ComplMap</a> <font color="Maroon">L</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="yellow_7.html#NK1" target="_self">opp</a> </span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:51_1_1_1"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="yellow_7.html#K1" target="_self">ComplMap</a> <font color="Maroon">L</font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="yellow_7.html#NK1" target="_self">opp</a> </span>)</span>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:51_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="yellow_7.html#NK1" target="_self">opp</a> </span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ;<br/>
<a NAME="E4:51_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="waybel_1.html#K7">'not'</a> <span class="p1">(<span class="default"><a href="waybel_1.html#K7">'not'</a> <font color="Maroon">x</font></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel_1.html#T90">WAYBEL_1:90</a></i>;<br/>
<a NAME="E5:51_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="yellow_7.html#K1" target="_self">ComplMap</a> <font color="Maroon">L</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="waybel_1.html#K7">'not'</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="yellow_7.html#T2" target="_self">Th2</a></i>;<br/>
<b>hence </b><a NAME="E6:51_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="yellow_7.html#K1" target="_self">ComplMap</a> <font color="Maroon">L</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_7.html#T4" target="_self">Th4</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/><a NAME="E1:51_1_1_2"/>
( <span class="p1">(<span class="default"><a href="yellow_7.html#K1" target="_self">ComplMap</a> <font color="Maroon">L</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="waybel_1.html#K7">'not'</a> <font color="Maroon">x</font> &amp; <span class="p1">(<span class="default"><a href="yellow_7.html#K1" target="_self">ComplMap</a> <font color="Maroon">L</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="waybel_1.html#K7">'not'</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="yellow_7.html#T2" target="_self">Th2</a></i>;<br/><a NAME="E2:51_1_1_2"/><b>then </b>
( <span class="p1">(<span class="default"><a href="yellow_7.html#K1" target="_self">ComplMap</a> <font color="Maroon">L</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_1.html#K7">'not'</a> <font color="Maroon">x</font></span>)</span> <a href="lattice3.html#K8">~</a>  &amp; <span class="p1">(<span class="default"><a href="yellow_7.html#K1" target="_self">ComplMap</a> <font color="Maroon">L</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_1.html#K7">'not'</a> <font color="Maroon">y</font></span>)</span> <a href="lattice3.html#K8">~</a>  )
 ;<br/><a NAME="E3:51_1_1_2"/><b>then </b>
( (  <a href="waybel_1.html#K7">'not'</a> <font color="Maroon">x</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a>  <a href="waybel_1.html#K7">'not'</a> <font color="Maroon">y</font> implies <span class="p1">(<span class="default"><a href="yellow_7.html#K1" target="_self">ComplMap</a> <font color="Maroon">L</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><a href="yellow_7.html#K1" target="_self">ComplMap</a> <font color="Maroon">L</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> ) &amp; ( <span class="p1">(<span class="default"><a href="yellow_7.html#K1" target="_self">ComplMap</a> <font color="Maroon">L</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><a href="yellow_7.html#K1" target="_self">ComplMap</a> <font color="Maroon">L</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> implies  <a href="waybel_1.html#K7">'not'</a> <font color="Maroon">x</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a>  <a href="waybel_1.html#K7">'not'</a> <font color="Maroon">y</font> ) &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="waybel_1.html#K7">'not'</a> <span class="p1">(<span class="default"><a href="waybel_1.html#K7">'not'</a> <font color="Maroon">x</font></span>)</span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="waybel_1.html#K7">'not'</a> <span class="p1">(<span class="default"><a href="waybel_1.html#K7">'not'</a> <font color="Maroon">y</font></span>)</span> )
 <b>by </b><i><a class="ref" href="lattice3.html#T9">LATTICE3:9</a>, <a class="ref" href="waybel_1.html#T90">WAYBEL_1:90</a></i>;<br/><b>hence </b><a NAME="E4:51_1_1_2"/>
( <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font> iff <span class="p1">(<span class="default"><a href="yellow_7.html#K1" target="_self">ComplMap</a> <font color="Maroon">L</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><a href="yellow_7.html#K1" target="_self">ComplMap</a> <font color="Maroon">L</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="waybel_1.html#T86">WAYBEL_1:86</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:51_1_1"/>
 <a href="yellow_7.html#K1" target="_self">ComplMap</a> <font color="Maroon">L</font> is <a href="waybel_0.html#V23">isomorphic</a>
 <b>by </b><i><a class="txt" href="yellow_7.html#E1:51_1_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="waybel_0.html#T66">WAYBEL_0:66</a></i>;<br/>


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