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<b>let </b><font color="Maroon" title="c1">A</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="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">a1</font>, <font color="Olive" title="b2">a2</font>, <font color="Olive" title="b3">a3</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font>  st ( ( <font color="Olive" title="b1">a1</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Olive" title="b2">a2</font> &amp; <font color="Olive" title="b2">a2</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b3">a3</font> ) or ( <font color="Olive" title="b1">a1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b2">a2</font> &amp; <font color="Olive" title="b2">a2</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Olive" title="b3">a3</font> ) ) holds <br/><font color="Olive" title="b1">a1</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Olive" title="b3">a3</font></span><br/><b>let </b><font color="Maroon" title="c2">a1</font>, <font color="Maroon" title="c3">a2</font>, <font color="Maroon" title="c4">a3</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( ( <font color="Maroon" title="c2">a1</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c3">a2</font> &amp; <font color="Maroon" title="c3">a2</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c4">a3</font> ) or ( <font color="Maroon" title="c2">a1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c3">a2</font> &amp; <font color="Maroon" title="c3">a2</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c4">a3</font> ) ) implies <font color="Maroon" title="c2">a1</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c4">a3</font> )</span><br/>



<b>assume </b><a NAME="E1:25"/><i><font color="Green" title="E15">A1</font></i>: 
( ( <font color="Maroon" title="c2">a1</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c3">a2</font> &amp; <font color="Maroon" title="c3">a2</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c4">a3</font> ) or ( <font color="Maroon" title="c2">a1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c3">a2</font> &amp; <font color="Maroon" title="c3">a2</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c4">a3</font> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">a1</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c4">a3</font></span><br/>

<div><i><font color="Green" title="E16">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:25_1"/><i><font color="Green" title="E17">A3</font></i>: 
( <font color="Maroon" title="c2">a1</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c3">a2</font> &amp; <font color="Maroon" title="c3">a2</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c4">a3</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">a1</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c4">a3</font></span><br/><a NAME="E2:25_1"/><b>then </b><i><font color="Green" title="E18">A4</font></i>: 
( <font color="Maroon" title="c2">a1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c3">a2</font> &amp; <font color="Maroon" title="c3">a2</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c4">a3</font> )
 <b>by </b><i><a class="ref" href="orders_2.html#D10" target="_self" title="ORDERS_2:def.10">Def10</a></i>;<br/><a NAME="E3:25_1"/><b>then </b><i><font color="Green" title="E19">A5</font></i>: 
<font color="Maroon" title="c2">a1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c4">a3</font>
 <b>by </b><i><a class="ref" href="orders_2.html#T26" target="_self" title="ORDERS_2:th.26">Th26</a></i>;<br/><a NAME="E4:25_1"/>
<font color="Maroon" title="c2">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">a3</font>
 <b>by </b><i><a class="txt" href="#E1:25_1"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="#E2:25_1"><i><font color="Green" title="E18">A4</font></i></a>, <a class="ref" href="orders_2.html#T25" target="_self" title="ORDERS_2:th.25">Th25</a></i>;<br/><b>hence </b><a NAME="E5:25_1"/>
<font color="Maroon" title="c2">a1</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c4">a3</font>
 <b>by </b><i><a class="txt" href="#E3:25_1"><i><font color="Green" title="E19">A5</font></i></a>, <a class="ref" href="orders_2.html#D10" target="_self" title="ORDERS_2:def.10">Def10</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>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:25_2"/><i><font color="Green" title="E17">A6</font></i>: 
( <font color="Maroon" title="c2">a1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c3">a2</font> &amp; <font color="Maroon" title="c3">a2</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c4">a3</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">a1</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c4">a3</font></span><br/><a NAME="E2:25_2"/><b>then </b><i><font color="Green" title="E18">A7</font></i>: 
( <font color="Maroon" title="c2">a1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c3">a2</font> &amp; <font color="Maroon" title="c3">a2</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c4">a3</font> )
 <b>by </b><i><a class="ref" href="orders_2.html#D10" target="_self" title="ORDERS_2:def.10">Def10</a></i>;<br/><a NAME="E3:25_2"/><b>then </b><i><font color="Green" title="E19">A8</font></i>: 
<font color="Maroon" title="c2">a1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c4">a3</font>
 <b>by </b><i><a class="ref" href="orders_2.html#T26" target="_self" title="ORDERS_2:th.26">Th26</a></i>;<br/><a NAME="E4:25_2"/>
<font color="Maroon" title="c2">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">a3</font>
 <b>by </b><i><a class="txt" href="#E1:25_2"><i><font color="Green" title="E17">A6</font></i></a>, <a class="txt" href="#E2:25_2"><i><font color="Green" title="E18">A7</font></i></a>, <a class="ref" href="orders_2.html#T25" target="_self" title="ORDERS_2:th.25">Th25</a></i>;<br/><b>hence </b><a NAME="E5:25_2"/>
<font color="Maroon" title="c2">a1</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c4">a3</font>
 <b>by </b><i><a class="txt" href="#E3:25_2"><i><font color="Green" title="E19">A8</font></i></a>, <a class="ref" href="orders_2.html#D10" target="_self" title="ORDERS_2:def.10">Def10</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="E4:25"/>
<font color="Maroon" title="c2">a1</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c4">a3</font>
 <b>by </b><i><a class="txt" href="#E1:25"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="#E2:25"><i><font color="Green" title="E16">A2</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/>


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