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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#V4">antisymmetric</a> <a href="lattice3.html#V3">complete</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/><b>let </b><font color="Maroon">F</font> be  <a href="funcop_1.html#V1">Function-yielding</a> <a href="funct_1.html#NM1">Function</a>;<br/>



<a NAME="E1:63"/><i><font color="Green">E32</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="waybel_5.html#K4">\//</a> <font color="Maroon">F</font>,<font color="Maroon">L</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="waybel_5.html#K5">/\\</a> <font color="Maroon">F</font>,<span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="yellow_7.html#NK1" target="_self">opp</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:63_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font>
 ;<br/><a NAME="E2:63_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="waybel_5.html#K4">\//</a> <font color="Maroon">F</font>,<font color="Maroon">L</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="yellow_2.html#K4">\\/</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>,<font color="Maroon">L</font> &amp; <span class="p1">(<span class="default"><a href="waybel_5.html#K5">/\\</a> <font color="Maroon">F</font>,<span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="yellow_7.html#NK1" target="_self">opp</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="yellow_2.html#K5">//\</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="yellow_7.html#NK1" target="_self">opp</a> </span>)</span> )
 <b>by </b><i><a class="ref" href="waybel_5.html#D1">WAYBEL_5:def 1</a>, <a class="ref" href="waybel_5.html#D2">WAYBEL_5:def 2</a></i>;<br/><b>hence </b><a NAME="E3:63_1"/>
<span class="p1">(<span class="default"><a href="waybel_5.html#K4">\//</a> <font color="Maroon">F</font>,<font color="Maroon">L</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_5.html#K5">/\\</a> <font color="Maroon">F</font>,<span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="yellow_7.html#NK1" target="_self">opp</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="yellow_7.html#T6" target="_self">Th6</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:63"/>
 <a href="waybel_5.html#K4">\//</a> <font color="Maroon">F</font>,<font color="Maroon">L</font> <a href="funct_2.html#R1">=</a>  <a href="waybel_5.html#K5">/\\</a> <font color="Maroon">F</font>,<span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="yellow_7.html#NK1" target="_self">opp</a> </span>)</span>
 <b>by </b><i><a class="ref" href="yellow_7.html#T7" target="_self">Th7</a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>

<a NAME="E4:63"/><i><font color="Green">E33</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="waybel_5.html#K5">/\\</a> <font color="Maroon">F</font>,<font color="Maroon">L</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="waybel_5.html#K4">\//</a> <font color="Maroon">F</font>,<span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="yellow_7.html#NK1" target="_self">opp</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:63_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font>
 ;<br/><a NAME="E2:63_2"/><b>then </b>
( <span class="p1">(<span class="default"><a href="waybel_5.html#K5">/\\</a> <font color="Maroon">F</font>,<font color="Maroon">L</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="yellow_2.html#K5">//\</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>,<font color="Maroon">L</font> &amp; <span class="p1">(<span class="default"><a href="waybel_5.html#K4">\//</a> <font color="Maroon">F</font>,<span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="yellow_7.html#NK1" target="_self">opp</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="yellow_2.html#K4">\\/</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="yellow_7.html#NK1" target="_self">opp</a> </span>)</span> )
 <b>by </b><i><a class="ref" href="waybel_5.html#D1">WAYBEL_5:def 1</a>, <a class="ref" href="waybel_5.html#D2">WAYBEL_5:def 2</a></i>;<br/><b>hence </b><a NAME="E3:63_2"/>
<span class="p1">(<span class="default"><a href="waybel_5.html#K5">/\\</a> <font color="Maroon">F</font>,<font color="Maroon">L</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_5.html#K4">\//</a> <font color="Maroon">F</font>,<span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="yellow_7.html#NK1" target="_self">opp</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="yellow_7.html#T6" target="_self">Th6</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:63"/>
 <a href="waybel_5.html#K5">/\\</a> <font color="Maroon">F</font>,<font color="Maroon">L</font> <a href="funct_2.html#R1">=</a>  <a href="waybel_5.html#K4">\//</a> <font color="Maroon">F</font>,<span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="yellow_7.html#NK1" target="_self">opp</a> </span>)</span>
 <b>by </b><i><a class="ref" href="yellow_7.html#T8" target="_self">Th8</a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


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