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<b>let </b><font color="Maroon" title="c1">L</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="orders_2.html#V5" title="ORDERS_2:attr.5">antisymmetric</a> <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">F</font> being <a href="funcop_1.html#V1" title="FUNCOP_1:attr.1">Function-yielding</a> <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> holds <br/> (  <a href="waybel_5.html#K4" title="WAYBEL_5:func.4">\//</a> <font color="Olive" title="b1">F</font>,<font color="Maroon" title="c1">L</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="waybel_5.html#K5" title="WAYBEL_5:func.5">/\\</a> <font color="Olive" title="b1">F</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a> </span>)</span> &amp;  <a href="waybel_5.html#K5" title="WAYBEL_5:func.5">/\\</a> <font color="Olive" title="b1">F</font>,<font color="Maroon" title="c1">L</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="waybel_5.html#K4" title="WAYBEL_5:func.4">\//</a> <font color="Olive" title="b1">F</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a> </span>)</span> )</span><br/><b>let </b><font color="Maroon" title="c2">F</font> be  <a href="funcop_1.html#V1" title="FUNCOP_1:attr.1">Function-yielding</a> <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="waybel_5.html#K4" title="WAYBEL_5:func.4">\//</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c1">L</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="waybel_5.html#K5" title="WAYBEL_5:func.5">/\\</a> <font color="Maroon" title="c2">F</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a> </span>)</span> &amp;  <a href="waybel_5.html#K5" title="WAYBEL_5:func.5">/\\</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c1">L</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="waybel_5.html#K4" title="WAYBEL_5:func.4">\//</a> <font color="Maroon" title="c2">F</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a> </span>)</span> )</span><br/>



<a NAME="E1:60"/><i><font color="Green" title="E37">A1</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="waybel_5.html#K4" title="WAYBEL_5:func.4">\//</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">F</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="waybel_5.html#K5" title="WAYBEL_5:func.5">/\\</a> <font color="Maroon" title="c2">F</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">F</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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">F</font> implies <span class="p1">(<span class="default"><a href="waybel_5.html#K4" title="WAYBEL_5:func.4">\//</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="waybel_5.html#K5" title="WAYBEL_5:func.5">/\\</a> <font color="Maroon" title="c2">F</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font> )</span><br/><b>assume </b><a NAME="E1:60_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="waybel_5.html#K4" title="WAYBEL_5:func.4">\//</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="waybel_5.html#K5" title="WAYBEL_5:func.5">/\\</a> <font color="Maroon" title="c2">F</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font></span><br/><a NAME="E2:60_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="waybel_5.html#K4" title="WAYBEL_5:func.4">\//</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_2.html#K4" title="YELLOW_2:func.4">\\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font></span>)</span>,<font color="Maroon" title="c1">L</font> &amp; <span class="p1">(<span class="default"><a href="waybel_5.html#K5" title="WAYBEL_5:func.5">/\\</a> <font color="Maroon" title="c2">F</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_2.html#K5" title="YELLOW_2:func.5">//\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a> </span>)</span> )
 <b>by </b><i><a class="ref" href="waybel_5.html#D1" title="WAYBEL_5:def.1">WAYBEL_5:def 1</a>, <a class="ref" href="waybel_5.html#D2" title="WAYBEL_5:def.2">WAYBEL_5:def 2</a></i>;<br/><b>hence </b><a NAME="E3:60_1"/>
<span class="p1">(<span class="default"><a href="waybel_5.html#K4" title="WAYBEL_5:func.4">\//</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="waybel_5.html#K5" title="WAYBEL_5:func.5">/\\</a> <font color="Maroon" title="c2">F</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font>
 <b>by </b><i><a class="ref" href="yellow_7.html#T49" target="_self" title="YELLOW_7:th.49">Th49</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:60"/>
 <a href="waybel_5.html#K4" title="WAYBEL_5:func.4">\//</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c1">L</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="waybel_5.html#K5" title="WAYBEL_5:func.5">/\\</a> <font color="Maroon" title="c2">F</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a> </span>)</span>
 <b>by </b><i><a class="txt" href="#E1:60"><i><font color="Green" title="E37">A1</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="waybel_5.html#K5" title="WAYBEL_5:func.5">/\\</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c1">L</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="waybel_5.html#K4" title="WAYBEL_5:func.4">\//</a> <font color="Maroon" title="c2">F</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a> </span>)</span></span><br/>

<a NAME="E4:60"/><i><font color="Green" title="E38">A2</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="waybel_5.html#K5" title="WAYBEL_5:func.5">/\\</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">F</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="waybel_5.html#K4" title="WAYBEL_5:func.4">\//</a> <font color="Maroon" title="c2">F</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">F</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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">F</font> implies <span class="p1">(<span class="default"><a href="waybel_5.html#K5" title="WAYBEL_5:func.5">/\\</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="waybel_5.html#K4" title="WAYBEL_5:func.4">\//</a> <font color="Maroon" title="c2">F</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font> )</span><br/><b>assume </b><a NAME="E1:60_2"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="waybel_5.html#K5" title="WAYBEL_5:func.5">/\\</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="waybel_5.html#K4" title="WAYBEL_5:func.4">\//</a> <font color="Maroon" title="c2">F</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font></span><br/><a NAME="E2:60_2"/><b>then </b>
( <span class="p1">(<span class="default"><a href="waybel_5.html#K5" title="WAYBEL_5:func.5">/\\</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_2.html#K5" title="YELLOW_2:func.5">//\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font></span>)</span>,<font color="Maroon" title="c1">L</font> &amp; <span class="p1">(<span class="default"><a href="waybel_5.html#K4" title="WAYBEL_5:func.4">\//</a> <font color="Maroon" title="c2">F</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_2.html#K4" title="YELLOW_2:func.4">\\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a> </span>)</span> )
 <b>by </b><i><a class="ref" href="waybel_5.html#D1" title="WAYBEL_5:def.1">WAYBEL_5:def 1</a>, <a class="ref" href="waybel_5.html#D2" title="WAYBEL_5:def.2">WAYBEL_5:def 2</a></i>;<br/><b>hence </b><a NAME="E3:60_2"/>
<span class="p1">(<span class="default"><a href="waybel_5.html#K5" title="WAYBEL_5:func.5">/\\</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="waybel_5.html#K4" title="WAYBEL_5:func.4">\//</a> <font color="Maroon" title="c2">F</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font>
 <b>by </b><i><a class="ref" href="yellow_7.html#T49" target="_self" title="YELLOW_7:th.49">Th49</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:60"/>
 <a href="waybel_5.html#K5" title="WAYBEL_5:func.5">/\\</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c1">L</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="waybel_5.html#K4" title="WAYBEL_5:func.4">\//</a> <font color="Maroon" title="c2">F</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a> </span>)</span>
 <b>by </b><i><a class="txt" href="#E4:60"><i><font color="Green" title="E38">A2</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
