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<b>let </b><font color="Maroon" title="c1">N</font> be  <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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">n1</font>, <font color="Olive" title="b2">n2</font>, <font color="Olive" title="b3">n3</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">N</font>  st <font color="Maroon" title="c1">N</font> is <a href="orders_2.html#V4" title="ORDERS_2:attr.4">transitive</a> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b1">n1</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Olive" title="b2">n2</font></span>)</span> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Olive" title="b3">n3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n1</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">n2</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Olive" title="b3">n3</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">n1</font>, <font color="Maroon" title="c3">n2</font>, <font color="Maroon" title="c4">n3</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">N</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">N</font> is <a href="orders_2.html#V4" title="ORDERS_2:attr.4">transitive</a> implies <span class="p1">(<span class="default"><font color="Maroon" title="c2">n1</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Maroon" title="c3">n2</font></span>)</span> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Maroon" title="c4">n3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n1</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n2</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Maroon" title="c4">n3</font></span>)</span> )</span><br/>







<b>assume </b><a NAME="E1:41"/><i><font color="Green" title="E22">A1</font></i>: 
<font color="Maroon" title="c1">N</font> is <a href="orders_2.html#V4" title="ORDERS_2:attr.4">transitive</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n1</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Maroon" title="c3">n2</font></span>)</span> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Maroon" title="c4">n3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n1</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n2</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Maroon" title="c4">n3</font></span>)</span></span><br/>

<a NAME="E2:41"/><i><font color="Green" title="E23">A2</font></i>: 
( <font color="Maroon" title="c2">n1</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Maroon" title="c3">n2</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">n1</font> &amp; <font color="Maroon" title="c2">n1</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Maroon" title="c3">n2</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c3">n2</font> &amp; <font color="Maroon" title="c3">n2</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Maroon" title="c4">n3</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c3">n2</font> &amp; <font color="Maroon" title="c3">n2</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Maroon" title="c4">n3</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c4">n3</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">n1</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Maroon" title="c3">n2</font></span>)</span> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Maroon" title="c4">n3</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">n1</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Maroon" title="c3">n2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">n1</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Maroon" title="c3">n2</font></span>)</span> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Maroon" title="c4">n3</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c4">n3</font> )
 
<b>by </b><i><a class="txt" href="lattice3.html#E28"><i><font color="Green" title="E17">Lm2</font></i></a></i>;<br/>
<a NAME="E3:41"/><b>then </b><i><font color="Green" title="E24">A3</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">n1</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Maroon" title="c3">n2</font></span>)</span> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Maroon" title="c4">n3</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">n1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">n1</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Maroon" title="c3">n2</font></span>)</span> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Maroon" title="c4">n3</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c3">n2</font> )
 
<b>by </b><i><a class="txt" href="lattice3.html#E1:41"><i><font color="Green" title="E22">A1</font></i></a>, <a class="ref" href="orders_2.html#T26" title="ORDERS_2:th.26">ORDERS_2:26</a></i>;<br/>
<a NAME="E4:41"/><b>then </b><i><font color="Green" title="E25">A4</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">n1</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Maroon" title="c3">n2</font></span>)</span> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Maroon" title="c4">n3</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c3">n2</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Maroon" title="c4">n3</font>
 
<b>by </b><i><a class="txt" href="lattice3.html#E2:41"><i><font color="Green" title="E23">A2</font></i></a>, <a class="txt" href="lattice3.html#E28"><i><font color="Green" title="E17">Lm2</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">n4</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">N</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">n4</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">n1</font> &amp; <font color="Maroon" title="c5">n4</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c3">n2</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Maroon" title="c4">n3</font> implies <font color="Maroon" title="c5">n4</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n1</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Maroon" title="c3">n2</font></span>)</span> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Maroon" title="c4">n3</font> )</span><br/><b>assume </b><a NAME="E1:41_1"/><i><font color="Green" title="E26">A5</font></i>: 
( <font color="Maroon" title="c5">n4</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">n1</font> &amp; <font color="Maroon" title="c5">n4</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c3">n2</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Maroon" title="c4">n3</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">n4</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n1</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Maroon" title="c3">n2</font></span>)</span> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Maroon" title="c4">n3</font></span><br/><a NAME="E2:41_1"/><b>then </b><i><font color="Green" title="E27">A6</font></i>: 
( <font color="Maroon" title="c5">n4</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c3">n2</font> &amp; <font color="Maroon" title="c5">n4</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c4">n3</font> )
 <b>by </b><i><a class="txt" href="lattice3.html#E1:41"><i><font color="Green" title="E22">A1</font></i></a>, <a class="txt" href="lattice3.html#E2:41"><i><font color="Green" title="E23">A2</font></i></a>, <a class="ref" href="orders_2.html#T26" title="ORDERS_2:th.26">ORDERS_2:26</a></i>;<br/><a NAME="E3:41_1"/><b>then </b>
<font color="Maroon" title="c5">n4</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">n1</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Maroon" title="c3">n2</font>
 <b>by </b><i><a class="txt" href="lattice3.html#E1:41_1"><i><font color="Green" title="E26">A5</font></i></a>, <a class="txt" href="lattice3.html#E28"><i><font color="Green" title="E17">Lm2</font></i></a></i>;<br/><b>hence </b><a NAME="E4:41_1"/>
<font color="Maroon" title="c5">n4</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n1</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Maroon" title="c3">n2</font></span>)</span> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Maroon" title="c4">n3</font>
 <b>by </b><i><a class="txt" href="lattice3.html#E2:41_1"><i><font color="Green" title="E27">A6</font></i></a>, <a class="txt" href="lattice3.html#E28"><i><font color="Green" title="E17">Lm2</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:41"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">n1</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Maroon" title="c3">n2</font></span>)</span> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Maroon" title="c4">n3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n1</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n2</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Maroon" title="c4">n3</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice3.html#E3:41"><i><font color="Green" title="E24">A3</font></i></a>, <a class="txt" href="lattice3.html#E4:41"><i><font color="Green" title="E25">A4</font></i></a>, <a class="ref" href="lattice3.html#D14" target="_self" title="LATTICE3:def.14">Def14</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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