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<span class="kw">let </span><font color="Maroon" title="c1">R</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="orders_2.html#V3" title="ORDERS_2:attr.3">reflexive</a>   <a href="orders_2.html#V4" title="ORDERS_2:attr.4">transitive</a>   <a href="orders_2.html#V5" title="ORDERS_2:attr.5">antisymmetric</a>   <a href="waybel_0.html#V24" title="WAYBEL_0:attr.24">up-complete</a>   <a href="waybel_9.html#L1" title="WAYBEL_9:struct.1">TopRelStr</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">S</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">R</font><br/>  for <font color="Olive" title="b2">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>  st <font color="Olive" title="b2">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">S</font> holds <br/><font color="Olive" title="b2">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> (<font color="Olive" title="b1">S</font>,<font color="Maroon" title="c1">R</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c2">S</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">R</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">  for <font color="Olive" title="b1">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>  st <font color="Olive" title="b1">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">S</font> holds <br/><font color="Olive" title="b1">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> (<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">R</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c3">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</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">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">S</font> implies <font color="Maroon" title="c3">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> (<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">R</font>) )</span><br/>





<span class="kw">assume </span><a NAME="E1:25"/><span class="lab"><font color="Green" title="E14">A1</font></span>: 
<font color="Maroon" title="c3">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">S</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">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> (<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">R</font>)</span><br/>

<a NAME="E2:25"/>
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_0.html#T75" onmouseover="rs('waybel_0/T75')" onmouseout="rh()">WAYBEL_0:75</a></span>;<br/>
<a NAME="E3:25"/><span class="kw">then </span>
<font color="Maroon" title="c2">S</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a>  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> (<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">R</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T30" onmouseover="rs('yellow_0/T30')" onmouseout="rh()">YELLOW_0:30</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:25"/>
<font color="Maroon" title="c3">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> (<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">R</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:25"><span class="lab"><font color="Green" title="E14">A1</font></span></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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