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<span class="kw">let </span><font color="Maroon" title="c1">L</font> be   <a href="orders_2.html#V3" title="ORDERS_2:attr.3">reflexive</a>   <a href="orders_2.html#V5" title="ORDERS_2:attr.5">antisymmetric</a>   <a href="lattice3.html#V1" title="LATTICE3:attr.1">with_suprema</a>   <a href="lattice3.html#V2" title="LATTICE3:attr.2">with_infima</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">a</font>, <font color="Olive" title="b2">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> holds <br/> ( <font color="Olive" title="b1">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> iff <font color="Olive" title="b1">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</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="c2">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> iff <font color="Maroon" title="c2">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> )</span><br/>



<span class="kw">thus </span><a NAME="E1:5"/>
( <font color="Maroon" title="c2">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> implies <font color="Maroon" title="c2">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</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="c2">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> implies <font color="Maroon" title="c2">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="5_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:5_1"/>
<font color="Maroon" title="c2">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</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="c2">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font></span><br/>

<a NAME="E2:5_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T23" onmouseover="rs('yellow_0/T23')" onmouseout="rh()">YELLOW_0:23</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:5_1"/>
<font color="Maroon" title="c2">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T24" onmouseover="rs('yellow_0/T24')" onmouseout="rh()">YELLOW_0:24</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>

<span class="kw">assume </span><a NAME="E2:5"/>
<font color="Maroon" title="c2">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</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="c2">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font></span><br/>

<a NAME="E3:5"/><span class="kw">then </span>
<font color="Maroon" title="c3">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T22" onmouseover="rs('yellow_0/T22')" onmouseout="rh()">YELLOW_0:22</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:5"/>
<font color="Maroon" title="c2">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T25" onmouseover="rs('yellow_0/T25')" onmouseout="rh()">YELLOW_0:25</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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