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<b>let </b><font color="Maroon" title="c1">L</font> be  <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="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">D1</font>, <font color="Maroon" title="c3">D2</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font>;<br/>



<a NAME="E1:73"/><i><font color="Green" title="E21">A1</font></i>: 
 <a href="waybel_0.html#K3" title="WAYBEL_0:func.3">downarrow</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">D1</font> <a href="yellow_4.html#K4" title="YELLOW_4:func.4">"/\"</a> <font color="Maroon" title="c3">D2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">s</font> where <font color="Olive" title="b1">s</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> :  ex <font color="Olive" title="b2">z</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">s</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b2">z</font> &amp; <font color="Olive" title="b2">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D1</font> <a href="yellow_4.html#K4" title="YELLOW_4:func.4">"/\"</a> <font color="Maroon" title="c3">D2</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="waybel_0.html#T14" title="WAYBEL_0:th.14">WAYBEL_0:14</a></i>;<br/>
<a NAME="E2:73"/><i><font color="Green" title="E22">A2</font></i>: 
 <a href="waybel_0.html#K3" title="WAYBEL_0:func.3">downarrow</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K3" title="WAYBEL_0:func.3">downarrow</a> <font color="Maroon" title="c2">D1</font></span>)</span> <a href="yellow_4.html#K4" title="YELLOW_4:func.4">"/\"</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K3" title="WAYBEL_0:func.3">downarrow</a> <font color="Maroon" title="c3">D2</font></span>)</span></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> where <font color="Olive" title="b1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> :  ex <font color="Olive" title="b2">t</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b2">t</font> &amp; <font color="Olive" title="b2">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K3" title="WAYBEL_0:func.3">downarrow</a> <font color="Maroon" title="c2">D1</font></span>)</span> <a href="yellow_4.html#K4" title="YELLOW_4:func.4">"/\"</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K3" title="WAYBEL_0:func.3">downarrow</a> <font color="Maroon" title="c3">D2</font></span>)</span> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="waybel_0.html#T14" title="WAYBEL_0:th.14">WAYBEL_0:14</a></i>;<br/>
<a NAME="E3:73"/><i><font color="Green" title="E23">A3</font></i>: 
 <a href="waybel_0.html#K3" title="WAYBEL_0:func.3">downarrow</a> <font color="Maroon" title="c2">D1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">s1</font> where <font color="Olive" title="b1">s1</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> :  ex <font color="Olive" title="b2">z</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">s1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b2">z</font> &amp; <font color="Olive" title="b2">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D1</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="waybel_0.html#T14" title="WAYBEL_0:th.14">WAYBEL_0:14</a></i>;<br/>
<a NAME="E4:73"/><i><font color="Green" title="E24">A4</font></i>: 
 <a href="waybel_0.html#K3" title="WAYBEL_0:func.3">downarrow</a> <font color="Maroon" title="c3">D2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">s2</font> where <font color="Olive" title="b1">s2</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> :  ex <font color="Olive" title="b2">z</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">s2</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b2">z</font> &amp; <font color="Olive" title="b2">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D2</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="waybel_0.html#T14" title="WAYBEL_0:th.14">WAYBEL_0:14</a></i>;<br/>
<b>let </b><font color="Maroon" title="c4">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E5:73"/>
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K3" title="WAYBEL_0:func.3">downarrow</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K3" title="WAYBEL_0:func.3">downarrow</a> <font color="Maroon" title="c2">D1</font></span>)</span> <a href="yellow_4.html#K4" title="YELLOW_4:func.4">"/\"</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K3" title="WAYBEL_0:func.3">downarrow</a> <font color="Maroon" title="c3">D2</font></span>)</span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E7:73"/><i><font color="Green" title="E25">A5</font></i>: 
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font>
 <b>and </b><br/><a NAME="E8:73"/><i><font color="Green" title="E26">A6</font></i>: 
 ex <font color="Olive" title="b1">t</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Maroon" title="c5">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b1">t</font> &amp; <font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K3" title="WAYBEL_0:func.3">downarrow</a> <font color="Maroon" title="c2">D1</font></span>)</span> <a href="yellow_4.html#K4" title="YELLOW_4:func.4">"/\"</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K3" title="WAYBEL_0:func.3">downarrow</a> <font color="Maroon" title="c3">D2</font></span>)</span> )
 <b>by </b><i><a class="txt" href="yellow_4.html#E2:73"><i><font color="Green" title="E22">A2</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">x1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E10:73"/><i><font color="Green" title="E27">A7</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c6">x1</font>
 <b>and </b><br/><a NAME="E11:73"/><i><font color="Green" title="E28">A8</font></i>: 
<font color="Maroon" title="c6">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K3" title="WAYBEL_0:func.3">downarrow</a> <font color="Maroon" title="c2">D1</font></span>)</span> <a href="yellow_4.html#K4" title="YELLOW_4:func.4">"/\"</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K3" title="WAYBEL_0:func.3">downarrow</a> <font color="Maroon" title="c3">D2</font></span>)</span>
 <b>by </b><i><a class="txt" href="yellow_4.html#E8:73"><i><font color="Green" title="E26">A6</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">a</font>, <font color="Maroon" title="c8">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E13:73"/><i><font color="Green" title="E29">A9</font></i>: 
<font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c8">b</font>
 <b>and </b><br/><a NAME="E14:73"/><i><font color="Green" title="E30">A10</font></i>: 
( <font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K3" title="WAYBEL_0:func.3">downarrow</a> <font color="Maroon" title="c2">D1</font> &amp; <font color="Maroon" title="c8">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K3" title="WAYBEL_0:func.3">downarrow</a> <font color="Maroon" title="c3">D2</font> )
 <b>by </b><i><a class="txt" href="yellow_4.html#E11:73"><i><font color="Green" title="E28">A8</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">A1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E16:73"/><i><font color="Green" title="E31">A11</font></i>: 
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">A1</font>
 <b>and </b><br/><a NAME="E17:73"/><i><font color="Green" title="E32">A12</font></i>: 
 ex <font color="Olive" title="b1">z</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Maroon" title="c9">A1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b1">z</font> &amp; <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D1</font> )
 <b>by </b><i><a class="txt" href="yellow_4.html#E3:73"><i><font color="Green" title="E23">A3</font></i></a>, <a class="txt" href="yellow_4.html#E14:73"><i><font color="Green" title="E30">A10</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">B1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E19:73"/><i><font color="Green" title="E33">A13</font></i>: 
<font color="Maroon" title="c8">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">B1</font>
 <b>and </b><br/><a NAME="E20:73"/><i><font color="Green" title="E34">A14</font></i>: 
 ex <font color="Olive" title="b1">z</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Maroon" title="c10">B1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b1">z</font> &amp; <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D2</font> )
 <b>by </b><i><a class="txt" href="yellow_4.html#E4:73"><i><font color="Green" title="E24">A4</font></i></a>, <a class="txt" href="yellow_4.html#E14:73"><i><font color="Green" title="E30">A10</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c11">a1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E22:73"/><i><font color="Green" title="E35">A15</font></i>: 
( <font color="Maroon" title="c9">A1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c11">a1</font> &amp; <font color="Maroon" title="c11">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D1</font> )
 <b>by </b><i><a class="txt" href="yellow_4.html#E17:73"><i><font color="Green" title="E32">A12</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">b1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E24:73"/><i><font color="Green" title="E36">A16</font></i>: 
( <font color="Maroon" title="c10">B1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c12">b1</font> &amp; <font color="Maroon" title="c12">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D2</font> )
 <b>by </b><i><a class="txt" href="yellow_4.html#E20:73"><i><font color="Green" title="E34">A14</font></i></a></i>;<br/>
<a NAME="E25:73"/>
<font color="Maroon" title="c6">x1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c11">a1</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c12">b1</font>
 
<b>by </b><i><a class="txt" href="yellow_4.html#E13:73"><i><font color="Green" title="E29">A9</font></i></a>, <a class="txt" href="yellow_4.html#E16:73"><i><font color="Green" title="E31">A11</font></i></a>, <a class="txt" href="yellow_4.html#E19:73"><i><font color="Green" title="E33">A13</font></i></a>, <a class="txt" href="yellow_4.html#E22:73"><i><font color="Green" title="E35">A15</font></i></a>, <a class="txt" href="yellow_4.html#E24:73"><i><font color="Green" title="E36">A16</font></i></a>, <a class="ref" href="yellow_3.html#T2" title="YELLOW_3:th.2">YELLOW_3:2</a></i>;<br/>
<a NAME="E26:73"/><b>then </b><i><font color="Green" title="E37">A17</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c11">a1</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c12">b1</font>
 
<b>by </b><i><a class="txt" href="yellow_4.html#E10:73"><i><font color="Green" title="E27">A7</font></i></a>, <a class="ref" href="orders_2.html#T26" title="ORDERS_2:th.26">ORDERS_2:26</a></i>;<br/>
<a NAME="E27:73"/>
<font color="Maroon" title="c11">a1</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c12">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D1</font> <a href="yellow_4.html#K4" title="YELLOW_4:func.4">"/\"</a> <font color="Maroon" title="c3">D2</font>
 
<b>by </b><i><a class="txt" href="yellow_4.html#E22:73"><i><font color="Green" title="E35">A15</font></i></a>, <a class="txt" href="yellow_4.html#E24:73"><i><font color="Green" title="E36">A16</font></i></a></i>;<br/>
<b>hence </b><a NAME="E28:73"/>
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K3" title="WAYBEL_0:func.3">downarrow</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">D1</font> <a href="yellow_4.html#K4" title="YELLOW_4:func.4">"/\"</a> <font color="Maroon" title="c3">D2</font></span>)</span>
 <b>by </b><i><a class="txt" href="yellow_4.html#E1:73"><i><font color="Green" title="E21">A1</font></i></a>, <a class="txt" href="yellow_4.html#E7:73"><i><font color="Green" title="E25">A5</font></i></a>, <a class="txt" href="yellow_4.html#E26:73"><i><font color="Green" title="E37">A17</font></i></a></i>;<br/>


</div>
