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<b>let </b><font color="Maroon" title="c1">L</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</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">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><br/> for <font color="Olive" title="b2">N</font> being  <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">L</font> holds  <font color="Olive" title="b1">a</font> <a href="waybel_2.html#K3" title="WAYBEL_2:func.3">"/\"</a> <font color="Olive" title="b2">N</font> is   <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">L</font></span><br/><b>let </b><font color="Maroon" title="c2">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <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">N</font> being  <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">L</font> holds  <font color="Maroon" title="c2">a</font> <a href="waybel_2.html#K3" title="WAYBEL_2:func.3">"/\"</a> <font color="Olive" title="b1">N</font> is   <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">L</font></span><br/><b>let </b><font color="Maroon" title="c3">N</font> be   <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">L</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">a</font> <a href="waybel_2.html#K3" title="WAYBEL_2:func.3">"/\"</a> <font color="Maroon" title="c3">N</font> is   <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">L</font></span><br/>





<b>set </b><font color="Maroon" title="c4">aN</font> = <font color="Maroon" title="c2">a</font> <a href="waybel_2.html#K3" title="WAYBEL_2:func.3">"/\"</a> <font color="Maroon" title="c3">N</font>;<br/>
<a NAME="E1:16"/>
<font color="Maroon" title="c2">a</font> <a href="waybel_2.html#K3" title="WAYBEL_2:func.3">"/\"</a> <font color="Maroon" title="c3">N</font> is <a href="orders_2.html#V4" title="ORDERS_2:attr.4">transitive</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c6">y</font>, <font color="Maroon" title="c7">z</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="waybel_2.html#K3" title="WAYBEL_2:func.3">"/\"</a> <font color="Maroon" title="c3">N</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_0.html#D2" title="YELLOW_0:def.2">YELLOW_0:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <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">y</font> or  not <font color="Maroon" title="c6">y</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c7">z</font> or <font color="Maroon" title="c5">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c7">z</font> )</span><br/>

<b>assume </b><a NAME="E1:16_1"/><i><font color="Green" title="E13">A1</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">y</font> &amp; <font color="Maroon" title="c6">y</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c7">z</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">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c7">z</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c8">x1</font> = <font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c9">y1</font> = <font color="Maroon" title="c6">y</font>, <font color="Maroon" title="c10">z1</font> = <font color="Maroon" title="c7">z</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">N</font> <b>by </b><i><a class="ref" href="waybel_2.html#T22" title="WAYBEL_2:th.22">WAYBEL_2:22</a></i>;<br/>
<a NAME="E3:16_1"/><i><font color="Green" title="E14">A2</font></i>: 
 <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">N</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c3">N</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="waybel_2.html#K3" title="WAYBEL_2:func.3">"/\"</a> <font color="Maroon" title="c3">N</font></span>)</span>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="waybel_2.html#K3" title="WAYBEL_2:func.3">"/\"</a> <font color="Maroon" title="c3">N</font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="waybel_2.html#D3" title="WAYBEL_2:def.3">WAYBEL_2:def 3</a></i>;<br/>
<a NAME="E4:16_1"/><b>then </b>
( <font color="Maroon" title="c8">x1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c9">y1</font> &amp; <font color="Maroon" title="c9">y1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c10">z1</font> )
 
<b>by </b><i><a class="txt" href="#E1:16_1"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="yellow_0.html#T1" title="YELLOW_0:th.1">YELLOW_0:1</a></i>;<br/>
<a NAME="E5:16_1"/><b>then </b>
<font color="Maroon" title="c8">x1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c10">z1</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D2" title="YELLOW_0:def.2">YELLOW_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E6:16_1"/>
<font color="Maroon" title="c5">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c7">z</font>
 <b>by </b><i><a class="txt" href="#E3:16_1"><i><font color="Green" title="E14">A2</font></i></a>, <a class="ref" href="yellow_0.html#T1" title="YELLOW_0:th.1">YELLOW_0:1</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="E2:16"/>
<font color="Maroon" title="c2">a</font> <a href="waybel_2.html#K3" title="WAYBEL_2:func.3">"/\"</a> <font color="Maroon" title="c3">N</font> is   <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">L</font>
 ; <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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