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<b>let </b><font color="Maroon" title="c1">L</font> be  <a href="orders_2.html#V3" title="ORDERS_2:attr.3">transitive</a> <a href="orders_2.html#V4" title="ORDERS_2:attr.4">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>, <font color="Maroon" title="c4">D3</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font>;<br/>



<b>thus </b><a NAME="E1:49"/>
<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="yellow_4.html#K4" title="YELLOW_4:func.4">"/\"</a> <font color="Maroon" title="c4">D3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">D1</font> <a href="yellow_4.html#K4" title="YELLOW_4:func.4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">D2</font> <a href="yellow_4.html#K4" title="YELLOW_4:func.4">"/\"</a> <font color="Maroon" title="c4">D3</font></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">q</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="E1:49_1"/>
<font color="Maroon" title="c5">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="yellow_4.html#K4" title="YELLOW_4:func.4">"/\"</a> <font color="Maroon" title="c4">D3</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c6">a1</font>, <font color="Maroon" title="c7">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="E3:49_1"/><i><font color="Green" title="E13">A1</font></i>: 
<font color="Maroon" title="c5">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a1</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c7">b1</font>
 <b>and </b><br/><a NAME="E4:49_1"/><i><font color="Green" title="E14">A2</font></i>: 
( <font color="Maroon" title="c6">a1</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> &amp; <font color="Maroon" title="c7">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">D3</font> )
 ;<br/>
<b>consider </b><font color="Maroon" title="c8">a11</font>, <font color="Maroon" title="c9">b11</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="E6:49_1"/><i><font color="Green" title="E15">A3</font></i>: 
<font color="Maroon" title="c6">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a11</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c9">b11</font>
 <b>and </b><br/><a NAME="E7:49_1"/><i><font color="Green" title="E16">A4</font></i>: 
( <font color="Maroon" title="c8">a11</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D1</font> &amp; <font color="Maroon" title="c9">b11</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#E4:49_1"><i><font color="Green" title="E14">A2</font></i></a></i>;<br/>
<a NAME="E8:49_1"/><i><font color="Green" title="E17">A5</font></i>: 
<font color="Maroon" title="c9">b11</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c7">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D2</font> <a href="yellow_4.html#K4" title="YELLOW_4:func.4">"/\"</a> <font color="Maroon" title="c4">D3</font>
 
<b>by </b><i><a class="txt" href="yellow_4.html#E4:49_1"><i><font color="Green" title="E14">A2</font></i></a>, <a class="txt" href="yellow_4.html#E7:49_1"><i><font color="Green" title="E16">A4</font></i></a></i>;<br/>
<a NAME="E9:49_1"/>
<font color="Maroon" title="c5">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a11</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">b11</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c7">b1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="yellow_4.html#E3:49_1"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="yellow_4.html#E6:49_1"><i><font color="Green" title="E15">A3</font></i></a>, <a class="ref" href="lattice3.html#T16" title="LATTICE3:th.16">LATTICE3:16</a></i>;<br/>
<b>hence </b><a NAME="E10:49_1"/>
<font color="Maroon" title="c5">q</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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">D2</font> <a href="yellow_4.html#K4" title="YELLOW_4:func.4">"/\"</a> <font color="Maroon" title="c4">D3</font></span>)</span>
 <b>by </b><i><a class="txt" href="yellow_4.html#E7:49_1"><i><font color="Green" title="E16">A4</font></i></a>, <a class="txt" href="yellow_4.html#E8:49_1"><i><font color="Green" title="E17">A5</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon" title="c5">q</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="E2:49"/>
<font color="Maroon" title="c5">q</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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">D2</font> <a href="yellow_4.html#K4" title="YELLOW_4:func.4">"/\"</a> <font color="Maroon" title="c4">D3</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c6">a1</font>, <font color="Maroon" title="c7">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="E4:49"/><i><font color="Green" title="E13">A6</font></i>: 
<font color="Maroon" title="c5">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a1</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c7">b1</font>
 <b>and </b><br/><a NAME="E5:49"/><i><font color="Green" title="E14">A7</font></i>: 
( <font color="Maroon" title="c6">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D1</font> &amp; <font color="Maroon" title="c7">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D2</font> <a href="yellow_4.html#K4" title="YELLOW_4:func.4">"/\"</a> <font color="Maroon" title="c4">D3</font> )
 ;<br/>
<b>consider </b><font color="Maroon" title="c8">a11</font>, <font color="Maroon" title="c9">b11</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="E7:49"/><i><font color="Green" title="E15">A8</font></i>: 
<font color="Maroon" title="c7">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a11</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c9">b11</font>
 <b>and </b><br/><a NAME="E8:49"/><i><font color="Green" title="E16">A9</font></i>: 
( <font color="Maroon" title="c8">a11</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D2</font> &amp; <font color="Maroon" title="c9">b11</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">D3</font> )
 <b>by </b><i><a class="txt" href="yellow_4.html#E5:49"><i><font color="Green" title="E14">A7</font></i></a></i>;<br/>
<a NAME="E9:49"/><i><font color="Green" title="E17">A10</font></i>: 
<font color="Maroon" title="c6">a1</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c8">a11</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#E5:49"><i><font color="Green" title="E14">A7</font></i></a>, <a class="txt" href="yellow_4.html#E8:49"><i><font color="Green" title="E16">A9</font></i></a></i>;<br/>
<a NAME="E10:49"/>
<font color="Maroon" title="c5">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">a1</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c8">a11</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c9">b11</font>
 
<b>by </b><i><a class="txt" href="yellow_4.html#E4:49"><i><font color="Green" title="E13">A6</font></i></a>, <a class="txt" href="yellow_4.html#E7:49"><i><font color="Green" title="E15">A8</font></i></a>, <a class="ref" href="lattice3.html#T16" title="LATTICE3:th.16">LATTICE3:16</a></i>;<br/>
<b>hence </b><a NAME="E11:49"/>
<font color="Maroon" title="c5">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="yellow_4.html#K4" title="YELLOW_4:func.4">"/\"</a> <font color="Maroon" title="c4">D3</font>
 <b>by </b><i><a class="txt" href="yellow_4.html#E8:49"><i><font color="Green" title="E16">A9</font></i></a>, <a class="txt" href="yellow_4.html#E9:49"><i><font color="Green" title="E17">A10</font></i></a></i>;<br/>


</div>
