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<span class="kw">let </span><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#L1" title="ORDERS_2:struct.1">RelStr</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">X</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st (  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Olive" title="b1">X</font>,<font color="Maroon" title="c1">L</font> or  <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <font color="Olive" title="b1">X</font>,<font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a>  ) holds <br/> <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> (<font color="Olive" title="b1">X</font>,<font color="Maroon" title="c1">L</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> (<font color="Olive" title="b1">X</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/><span class="kw">let </span><font color="Maroon" title="c2">X</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">L</font> or  <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a>  ) implies  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> (<font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">L</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> (<font color="Maroon" title="c2">X</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/>



<span class="kw">assume </span><a NAME="E1:16"/><span class="lab"><font color="Green" title="E12">A1</font></span>: 
(  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">L</font> or  <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a>  )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> (<font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">L</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> (<font color="Maroon" title="c2">X</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="E2:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E13">A2</font></span>: 
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_7.html#T10" target="_self" onmouseover="rs('yellow_7/T10')" onmouseout="rh()">Th10</a></span>;<br/>
<a NAME="E3:16"/><span class="kw">then </span>
 <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> (<font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">L</font>) <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a> <font color="Maroon" title="c2">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#D9" onmouseover="rs('yellow_0/D9')" onmouseout="rh()">YELLOW_0:def 9</a></span>;<br/>
<a NAME="E4:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E14">A3</font></span>: 
<span class="p1">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> (<font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a>  <a href="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a> <font color="Maroon" title="c2">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_7.html#T8" target="_self" onmouseover="rs('yellow_7/T8')" onmouseout="rh()">Th8</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E15">A4</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="16_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <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>  st <font color="Olive" title="b1">x</font> <a href="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a> <font color="Maroon" title="c2">X</font> holds <br/><font color="Olive" title="b1">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> (<font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a> </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <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>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">x</font> <a href="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a> <font color="Maroon" title="c2">X</font> implies <font color="Maroon" title="c3">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> (<font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a>  )</span><br/><span class="kw">assume </span><a NAME="E1:16_1"/>
<font color="Maroon" title="c3">x</font> <a href="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a> <font color="Maroon" title="c2">X</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> (<font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a> </span><br/><a NAME="E2:16_1"/><span class="kw">then </span>
 <a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c3">x</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a> <font color="Maroon" title="c2">X</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_7.html#T9" target="_self" onmouseover="rs('yellow_7/T9')" onmouseout="rh()">Th9</a></span>;<br/><a NAME="E3:16_1"/><span class="kw">then </span>
 <a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c3">x</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a>  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> (<font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">L</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16"><span class="lab"><font color="Green" title="E13">A2</font></span></a>, <a class="ref" href="yellow_0.html#D9" onmouseover="rs('yellow_0/D9')" onmouseout="rh()">YELLOW_0:def 9</a></span>;<br/><span class="kw">hence </span><a NAME="E4:16_1"/>
<font color="Maroon" title="c3">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> (<font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_7.html#T2" target="_self" onmouseover="rs('yellow_7/T2')" onmouseout="rh()">Th2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E6:16"/>
 <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="ref" href="yellow_7.html#T10" target="_self" onmouseover="rs('yellow_7/T10')" onmouseout="rh()">Th10</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:16"/>
 <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> (<font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">L</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> (<font color="Maroon" title="c2">X</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E4:16"><span class="lab"><font color="Green" title="E14">A3</font></span></a>, <a class="txt" href="#E5:16"><span class="lab"><font color="Green" title="E15">A4</font></span></a>, <a class="ref" href="yellow_0.html#D10" onmouseover="rs('yellow_0/D10')" onmouseout="rh()">YELLOW_0:def 10</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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