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<b>let </b><font color="Maroon">S</font> be  <a href="orders_2.html#V2">reflexive</a> <a href="orders_2.html#V4">antisymmetric</a> <a href="lattice3.html#V1">with_suprema</a> <a href="lattice3.html#V2">with_infima</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/><b>let </b><font color="Maroon">T</font> be  <a href="orders_2.html#V4">antisymmetric</a> <a href="lattice3.html#V1">with_suprema</a> <a href="lattice3.html#V2">with_infima</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/>



<b>assume </b><a NAME="E1:74"/><i><font color="Green">E18</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">T</font></span><a href="yellow_3.html#K3">:]</a></span> holds  <font color="Olive">x</font> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><font color="Olive">y</font> <a href="lattice3.html#K13">"\/"</a> <font color="Olive">z</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Olive">y</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Olive">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Olive">z</font></span>)</span>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_1.html#D3">WAYBEL_1:def 3</a><br/>

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font>, <font color="Maroon">z</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_1.html#D3">WAYBEL_1:def 3</a><br/>

<b>consider </b><font color="Maroon">s</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>;<br/>
<a NAME="E3:74"/>
<font color="Maroon">s</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">s</font>
 
;<br/>
<a NAME="E4:74"/><b>then </b><i><font color="Green">E20</font></i>: 
( <font color="Maroon">s</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> &amp; <font color="Maroon">s</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> )
 
<b>by </b><i><a class="ref" href="yellow_0.html#T24">YELLOW_0:24</a>, <a class="ref" href="yellow_0.html#T25">YELLOW_0:25</a></i>;<br/>
<b>thus </b><font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">z</font></span>)</span> = 
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">s</font>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">z</font></span>)</span></span>)</span></span><a href="yellow_3.html#K7">]</a></span> <a href="yellow_3.html#K9">`2</a> 
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">x</font></span><a href="yellow_3.html#K7">]</a></span> <a href="lattice3.html#K12">"/\"</a> <span class="p2"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">s</font>,<span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">z</font></span>)</span></span><a href="yellow_3.html#K7">]</a></span></span>)</span> <a href="yellow_3.html#K9">`2</a> 
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">x</font></span><a href="yellow_3.html#K7">]</a></span> <a href="lattice3.html#K12">"/\"</a> <span class="p2">(<span class="default"><span class="p3"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">y</font></span><a href="yellow_3.html#K7">]</a></span> <a href="lattice3.html#K13">"\/"</a> <span class="p3"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">z</font></span><a href="yellow_3.html#K7">]</a></span></span>)</span></span>)</span> <a href="yellow_3.html#K9">`2</a> 
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">x</font></span><a href="yellow_3.html#K7">]</a></span> <a href="lattice3.html#K12">"/\"</a> <span class="p3"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">y</font></span><a href="yellow_3.html#K7">]</a></span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><span class="p3"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">x</font></span><a href="yellow_3.html#K7">]</a></span> <a href="lattice3.html#K12">"/\"</a> <span class="p3"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">z</font></span><a href="yellow_3.html#K7">]</a></span></span>)</span></span>)</span> <a href="yellow_3.html#K9">`2</a> 
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">s</font>,<span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">y</font></span>)</span></span><a href="yellow_3.html#K7">]</a></span> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><span class="p3"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">x</font></span><a href="yellow_3.html#K7">]</a></span> <a href="lattice3.html#K12">"/\"</a> <span class="p3"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">z</font></span><a href="yellow_3.html#K7">]</a></span></span>)</span></span>)</span> <a href="yellow_3.html#K9">`2</a> 
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">s</font>,<span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">y</font></span>)</span></span><a href="yellow_3.html#K7">]</a></span> <a href="lattice3.html#K13">"\/"</a> <span class="p2"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">s</font>,<span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">z</font></span>)</span></span><a href="yellow_3.html#K7">]</a></span></span>)</span> <a href="yellow_3.html#K9">`2</a> 
<b>by </b><i>, </i>
<br/>.= 
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">y</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">z</font></span>)</span></span>)</span></span><a href="yellow_3.html#K7">]</a></span> <a href="yellow_3.html#K9">`2</a> 
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">y</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
;<br/>


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