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<b>let </b><font color="Maroon" title="c1">S</font> be  <a href="orders_2.html#V2" title="ORDERS_2:attr.2">reflexive</a> <a href="orders_2.html#V4" title="ORDERS_2:attr.4">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> ;<br/><b>let </b><font color="Maroon" title="c2">T</font> be  <a href="orders_2.html#V4" title="ORDERS_2:attr.4">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> ;<br/>



<b>assume </b><a NAME="E1:74"/><i><font color="Green" title="E29">A1</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span> holds  <font color="Olive" title="b1">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">y</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Olive" title="b3">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Olive" title="b3">z</font></span>)</span>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_1.html#D3" title="WAYBEL_1:def.3">WAYBEL_1:def 3</a><br/>

<b>let </b><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font>, <font color="Maroon" title="c5">z</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_1.html#D3" title="WAYBEL_1:def.3">WAYBEL_1:def 3</a><br/>

<b>consider </b><font color="Maroon" title="c6">s</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font>;<br/>
<a NAME="E3:74"/>
<font color="Maroon" title="c6">s</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">s</font>
 
;<br/>
<a NAME="E4:74"/><b>then </b><i><font color="Green" title="E30">A2</font></i>: 
( <font color="Maroon" title="c6">s</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c6">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">s</font> &amp; <font color="Maroon" title="c6">s</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c6">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">s</font> )
 
<b>by </b><i><a class="ref" href="yellow_0.html#T24" title="YELLOW_0:th.24">YELLOW_0:24</a>, <a class="ref" href="yellow_0.html#T25" title="YELLOW_0:th.25">YELLOW_0:25</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c3">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c5">z</font></span>)</span> = 
<span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c6">s</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c5">z</font></span>)</span></span>)</span></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="yellow_3.html#K9" title="YELLOW_3:func.9">`2</a> 
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c6">s</font>,<font color="Maroon" title="c3">x</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p2"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c6">s</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c5">z</font></span>)</span></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span></span>)</span> <a href="yellow_3.html#K9" title="YELLOW_3:func.9">`2</a> 
<b>by </b><i><a class="txt" href="yellow10.html#E4:74"><i><font color="Green" title="E30">A2</font></i></a>, <a class="ref" href="yellow10.html#T15" target="_self" title="YELLOW10:th.15">Th15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c6">s</font>,<font color="Maroon" title="c3">x</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p2">(<span class="default"><span class="p3"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c6">s</font>,<font color="Maroon" title="c4">y</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p3"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c6">s</font>,<font color="Maroon" title="c5">z</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span></span>)</span></span>)</span> <a href="yellow_3.html#K9" title="YELLOW_3:func.9">`2</a> 
<b>by </b><i><a class="txt" href="yellow10.html#E4:74"><i><font color="Green" title="E30">A2</font></i></a>, <a class="ref" href="yellow10.html#T16" target="_self" title="YELLOW10:th.16">Th16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c6">s</font>,<font color="Maroon" title="c3">x</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p3"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c6">s</font>,<font color="Maroon" title="c4">y</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><span class="p3"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c6">s</font>,<font color="Maroon" title="c3">x</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p3"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c6">s</font>,<font color="Maroon" title="c5">z</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span></span>)</span></span>)</span> <a href="yellow_3.html#K9" title="YELLOW_3:func.9">`2</a> 
<b>by </b><i><a class="txt" href="yellow10.html#E1:74"><i><font color="Green" title="E29">A1</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c6">s</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c4">y</font></span>)</span></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><span class="p3"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c6">s</font>,<font color="Maroon" title="c3">x</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p3"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c6">s</font>,<font color="Maroon" title="c5">z</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span></span>)</span></span>)</span> <a href="yellow_3.html#K9" title="YELLOW_3:func.9">`2</a> 
<b>by </b><i><a class="txt" href="yellow10.html#E4:74"><i><font color="Green" title="E30">A2</font></i></a>, <a class="ref" href="yellow10.html#T15" target="_self" title="YELLOW10:th.15">Th15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c6">s</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c4">y</font></span>)</span></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c6">s</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c5">z</font></span>)</span></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span></span>)</span> <a href="yellow_3.html#K9" title="YELLOW_3:func.9">`2</a> 
<b>by </b><i><a class="txt" href="yellow10.html#E4:74"><i><font color="Green" title="E30">A2</font></i></a>, <a class="ref" href="yellow10.html#T15" target="_self" title="YELLOW10:th.15">Th15</a></i>
<br/>.= 
<span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c6">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c5">z</font></span>)</span></span>)</span></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="yellow_3.html#K9" title="YELLOW_3:func.9">`2</a> 
<b>by </b><i><a class="txt" href="yellow10.html#E4:74"><i><font color="Green" title="E30">A2</font></i></a>, <a class="ref" href="yellow10.html#T16" target="_self" title="YELLOW10:th.16">Th16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c5">z</font></span>)</span>
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
;<br/>


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