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<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">V</font> be   <a href="orders_2.html#V5" title="ORDERS_2:attr.5">antisymmetric</a>   <a href="lattice3.html#V1" title="LATTICE3:attr.1">with_suprema</a>   <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">u1</font>, <font color="Olive" title="b2">u2</font>, <font color="Olive" title="b3">u3</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font>  st <font color="Maroon" title="c1">V</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">u1</font> <a href="lattice3.html#K10" title="LATTICE3:func.10">"\/"</a> <font color="Olive" title="b2">u2</font></span>)</span> <a href="lattice3.html#K10" title="LATTICE3:func.10">"\/"</a> <font color="Olive" title="b3">u3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">u1</font> <a href="lattice3.html#K10" title="LATTICE3:func.10">"\/"</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">u2</font> <a href="lattice3.html#K10" title="LATTICE3:func.10">"\/"</a> <font color="Olive" title="b3">u3</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">u1</font>, <font color="Maroon" title="c3">u2</font>, <font color="Maroon" title="c4">u3</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">V</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">u1</font> <a href="lattice3.html#K10" title="LATTICE3:func.10">"\/"</a> <font color="Maroon" title="c3">u2</font></span>)</span> <a href="lattice3.html#K10" title="LATTICE3:func.10">"\/"</a> <font color="Maroon" title="c4">u3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">u1</font> <a href="lattice3.html#K10" title="LATTICE3:func.10">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">u2</font> <a href="lattice3.html#K10" title="LATTICE3:func.10">"\/"</a> <font color="Maroon" title="c4">u3</font></span>)</span> )</span><br/>







<span class="kw">assume </span><a NAME="E1:39"/><span class="lab"><font color="Green" title="E20">A1</font></span>: 
<font color="Maroon" title="c1">V</font> is  <a href="orders_2.html#V4" title="ORDERS_2:attr.4">transitive</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">u1</font> <a href="lattice3.html#K10" title="LATTICE3:func.10">"\/"</a> <font color="Maroon" title="c3">u2</font></span>)</span> <a href="lattice3.html#K10" title="LATTICE3:func.10">"\/"</a> <font color="Maroon" title="c4">u3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">u1</font> <a href="lattice3.html#K10" title="LATTICE3:func.10">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">u2</font> <a href="lattice3.html#K10" title="LATTICE3:func.10">"\/"</a> <font color="Maroon" title="c4">u3</font></span>)</span></span><br/>

<a NAME="E2:39"/><span class="lab"><font color="Green" title="E21">A2</font></span>: 
<font color="Maroon" title="c2">u1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">u1</font> <a href="lattice3.html#K10" title="LATTICE3:func.10">"\/"</a> <font color="Maroon" title="c3">u2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="lattice3.html#E26"><span class="lab"><font color="Green" title="E15">Lm1</font></span></a></span>;<br/>
<a NAME="E3:39"/><span class="lab"><font color="Green" title="E22">A3</font></span>: 
<font color="Maroon" title="c3">u2</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">u1</font> <a href="lattice3.html#K10" title="LATTICE3:func.10">"\/"</a> <font color="Maroon" title="c3">u2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="lattice3.html#E26"><span class="lab"><font color="Green" title="E15">Lm1</font></span></a></span>;<br/>
<a NAME="E4:39"/><span class="lab"><font color="Green" title="E23">A4</font></span>: 
<font color="Maroon" title="c3">u2</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c3">u2</font> <a href="lattice3.html#K10" title="LATTICE3:func.10">"\/"</a> <font color="Maroon" title="c4">u3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="lattice3.html#E26"><span class="lab"><font color="Green" title="E15">Lm1</font></span></a></span>;<br/>
<a NAME="E5:39"/><span class="lab"><font color="Green" title="E24">A5</font></span>: 
<font color="Maroon" title="c4">u3</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c3">u2</font> <a href="lattice3.html#K10" title="LATTICE3:func.10">"\/"</a> <font color="Maroon" title="c4">u3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="lattice3.html#E26"><span class="lab"><font color="Green" title="E15">Lm1</font></span></a></span>;<br/>
<a NAME="E6:39"/><span class="lab"><font color="Green" title="E25">A6</font></span>: 
<font color="Maroon" title="c2">u1</font> <a href="lattice3.html#K10" title="LATTICE3:func.10">"\/"</a> <font color="Maroon" title="c3">u2</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">u1</font> <a href="lattice3.html#K10" title="LATTICE3:func.10">"\/"</a> <font color="Maroon" title="c3">u2</font></span>)</span> <a href="lattice3.html#K10" title="LATTICE3:func.10">"\/"</a> <font color="Maroon" title="c4">u3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="lattice3.html#E26"><span class="lab"><font color="Green" title="E15">Lm1</font></span></a></span>;<br/>
<a NAME="E7:39"/><span class="lab"><font color="Green" title="E26">A7</font></span>: 
<font color="Maroon" title="c4">u3</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">u1</font> <a href="lattice3.html#K10" title="LATTICE3:func.10">"\/"</a> <font color="Maroon" title="c3">u2</font></span>)</span> <a href="lattice3.html#K10" title="LATTICE3:func.10">"\/"</a> <font color="Maroon" title="c4">u3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="lattice3.html#E26"><span class="lab"><font color="Green" title="E15">Lm1</font></span></a></span>;<br/>
<a NAME="E8:39"/><span class="lab"><font color="Green" title="E27">A8</font></span>: 
<font color="Maroon" title="c2">u1</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">u1</font> <a href="lattice3.html#K10" title="LATTICE3:func.10">"\/"</a> <font color="Maroon" title="c3">u2</font></span>)</span> <a href="lattice3.html#K10" title="LATTICE3:func.10">"\/"</a> <font color="Maroon" title="c4">u3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E2:39"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="txt" href="#E6:39"><span class="lab"><font color="Green" title="E25">A6</font></span></a>, <a class="ref" href="orders_2.html#T3" title="ORDERS_2:th.3">ORDERS_2:3</a></span>;<br/>
<a NAME="E9:39"/>
<font color="Maroon" title="c3">u2</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">u1</font> <a href="lattice3.html#K10" title="LATTICE3:func.10">"\/"</a> <font color="Maroon" title="c3">u2</font></span>)</span> <a href="lattice3.html#K10" title="LATTICE3:func.10">"\/"</a> <font color="Maroon" title="c4">u3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E3:39"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="txt" href="#E6:39"><span class="lab"><font color="Green" title="E25">A6</font></span></a>, <a class="ref" href="orders_2.html#T3" title="ORDERS_2:th.3">ORDERS_2:3</a></span>;<br/>
<a NAME="E10:39"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A9</font></span>: 
<font color="Maroon" title="c3">u2</font> <a href="lattice3.html#K10" title="LATTICE3:func.10">"\/"</a> <font color="Maroon" title="c4">u3</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">u1</font> <a href="lattice3.html#K10" title="LATTICE3:func.10">"\/"</a> <font color="Maroon" title="c3">u2</font></span>)</span> <a href="lattice3.html#K10" title="LATTICE3:func.10">"\/"</a> <font color="Maroon" title="c4">u3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:39"><span class="lab"><font color="Green" title="E26">A7</font></span></a>, <a class="txt" href="lattice3.html#E26"><span class="lab"><font color="Green" title="E15">Lm1</font></span></a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">u4</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">u1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">u4</font> &amp; <font color="Maroon" title="c3">u2</font> <a href="lattice3.html#K10" title="LATTICE3:func.10">"\/"</a> <font color="Maroon" title="c4">u3</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">u4</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c2">u1</font> <a href="lattice3.html#K10" title="LATTICE3:func.10">"\/"</a> <font color="Maroon" title="c3">u2</font></span>)</span> <a href="lattice3.html#K10" title="LATTICE3:func.10">"\/"</a> <font color="Maroon" title="c4">u3</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">u4</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:39_1"/><span class="lab"><font color="Green" title="E29">A10</font></span>: 
<font color="Maroon" title="c2">u1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">u4</font>
 <span class="kw">and </span><br/><a NAME="E2:39_1"/><span class="lab"><font color="Green" title="E30">A11</font></span>: 
<font color="Maroon" title="c3">u2</font> <a href="lattice3.html#K10" title="LATTICE3:func.10">"\/"</a> <font color="Maroon" title="c4">u3</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">u4</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">u1</font> <a href="lattice3.html#K10" title="LATTICE3:func.10">"\/"</a> <font color="Maroon" title="c3">u2</font></span>)</span> <a href="lattice3.html#K10" title="LATTICE3:func.10">"\/"</a> <font color="Maroon" title="c4">u3</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">u4</font></span><br/><a NAME="E3:39_1"/><span class="lab"><font color="Green" title="E31">A12</font></span>: 
<font color="Maroon" title="c3">u2</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">u4</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E4:39"><span class="lab"><font color="Green" title="E23">A4</font></span></a>, <a class="txt" href="#E2:39_1"><span class="lab"><font color="Green" title="E30">A11</font></span></a>, <a class="ref" href="orders_2.html#T3" title="ORDERS_2:th.3">ORDERS_2:3</a></span>;<br/><a NAME="E4:39_1"/><span class="lab"><font color="Green" title="E32">A13</font></span>: 
<font color="Maroon" title="c4">u3</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">u4</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E5:39"><span class="lab"><font color="Green" title="E24">A5</font></span></a>, <a class="txt" href="#E2:39_1"><span class="lab"><font color="Green" title="E30">A11</font></span></a>, <a class="ref" href="orders_2.html#T3" title="ORDERS_2:th.3">ORDERS_2:3</a></span>;<br/><a NAME="E5:39_1"/>
<font color="Maroon" title="c2">u1</font> <a href="lattice3.html#K10" title="LATTICE3:func.10">"\/"</a> <font color="Maroon" title="c3">u2</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">u4</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_1"><span class="lab"><font color="Green" title="E29">A10</font></span></a>, <a class="txt" href="#E3:39_1"><span class="lab"><font color="Green" title="E31">A12</font></span></a>, <a class="txt" href="lattice3.html#E26"><span class="lab"><font color="Green" title="E15">Lm1</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E6:39_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">u1</font> <a href="lattice3.html#K10" title="LATTICE3:func.10">"\/"</a> <font color="Maroon" title="c3">u2</font></span>)</span> <a href="lattice3.html#K10" title="LATTICE3:func.10">"\/"</a> <font color="Maroon" title="c4">u3</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">u4</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:39_1"><span class="lab"><font color="Green" title="E32">A13</font></span></a>, <a class="txt" href="lattice3.html#E26"><span class="lab"><font color="Green" title="E15">Lm1</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E12:39"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">u1</font> <a href="lattice3.html#K10" title="LATTICE3:func.10">"\/"</a> <font color="Maroon" title="c3">u2</font></span>)</span> <a href="lattice3.html#K10" title="LATTICE3:func.10">"\/"</a> <font color="Maroon" title="c4">u3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">u1</font> <a href="lattice3.html#K10" title="LATTICE3:func.10">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">u2</font> <a href="lattice3.html#K10" title="LATTICE3:func.10">"\/"</a> <font color="Maroon" title="c4">u3</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:39"><span class="lab"><font color="Green" title="E27">A8</font></span></a>, <a class="txt" href="#E10:39"><span class="lab"><font color="Green" title="E28">A9</font></span></a>, <a class="ref" href="lattice3.html#D13" target="_self" title="LATTICE3:def.13">Def13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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