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<b>consider </b><font color="Maroon" title="c1">A</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>
<b>consider </b><font color="Maroon" title="c2">R</font> being   <a href="orders_1.html#NM2" title="ORDERS_1:NM.2">Order</a> of <font color="Maroon" title="c1">A</font>;<br/>
<b>take </b><font color="Maroon" title="c3">L</font> =  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">R</font> #); <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( not <font color="Maroon" title="c3">L</font> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> &amp; <font color="Maroon" title="c3">L</font> is <a href="orders_2.html#V3" title="ORDERS_2:attr.3">reflexive</a> &amp; <font color="Maroon" title="c3">L</font> is <a href="orders_2.html#V4" title="ORDERS_2:attr.4">transitive</a> &amp; <font color="Maroon" title="c3">L</font> is <a href="orders_2.html#V5" title="ORDERS_2:attr.5">antisymmetric</a> &amp; <font color="Maroon" title="c3">L</font> is <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a> &amp; <font color="Maroon" title="c3">L</font> is <a href="orders_2.html#V2" title="ORDERS_2:attr.2">total</a> )</span><br/>

<b>thus </b><a NAME="E3:4_1_1"/>
not the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">L</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="struct_0.html#D1" title="STRUCT_0:def.1">STRUCT_0:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">L</font> is <a href="orders_2.html#V3" title="ORDERS_2:attr.3">reflexive</a> &amp; <font color="Maroon" title="c3">L</font> is <a href="orders_2.html#V4" title="ORDERS_2:attr.4">transitive</a> &amp; <font color="Maroon" title="c3">L</font> is <a href="orders_2.html#V5" title="ORDERS_2:attr.5">antisymmetric</a> &amp; <font color="Maroon" title="c3">L</font> is <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a> &amp; <font color="Maroon" title="c3">L</font> is <a href="orders_2.html#V2" title="ORDERS_2:attr.2">total</a> )</span><br/>

<a NAME="E4:4_1_1"/><i><font color="Green" title="E5">A1</font></i>: 
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c2">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">L</font>
 
<b>by </b><i><a class="ref" href="orders_1.html#T97" title="ORDERS_1:th.97">ORDERS_1:97</a></i>;<br/>
<b>hence </b><a NAME="E5:4_1_1"/>
the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c3">L</font> <a href="relat_2.html#R1" title="RELAT_2:pred.1">is_reflexive_in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">L</font>
 <b>by </b><i><a class="ref" href="relat_2.html#D9" title="RELAT_2:def.9">RELAT_2:def 9</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="orders_2.html#D4" target="_self" title="ORDERS_2:def.4">ORDERS_2:def 4</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">L</font> is <a href="orders_2.html#V4" title="ORDERS_2:attr.4">transitive</a> &amp; <font color="Maroon" title="c3">L</font> is <a href="orders_2.html#V5" title="ORDERS_2:attr.5">antisymmetric</a> &amp; <font color="Maroon" title="c3">L</font> is <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a> &amp; <font color="Maroon" title="c3">L</font> is <a href="orders_2.html#V2" title="ORDERS_2:attr.2">total</a> )</span><br/>

<b>thus </b><a NAME="E6:4_1_1"/>
the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c3">L</font> <a href="relat_2.html#R8" title="RELAT_2:pred.8">is_transitive_in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">L</font>
 <b>by </b><i><a class="txt" href="orders_2.html#E4:4_1_1"><i><font color="Green" title="E5">A1</font></i></a>, <a class="ref" href="relat_2.html#D16" title="RELAT_2:def.16">RELAT_2:def 16</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="orders_2.html#D5" target="_self" title="ORDERS_2:def.5">ORDERS_2:def 5</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">L</font> is <a href="orders_2.html#V5" title="ORDERS_2:attr.5">antisymmetric</a> &amp; <font color="Maroon" title="c3">L</font> is <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a> &amp; <font color="Maroon" title="c3">L</font> is <a href="orders_2.html#V2" title="ORDERS_2:attr.2">total</a> )</span><br/>

<b>thus </b><a NAME="E7:4_1_1"/>
the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c3">L</font> <a href="relat_2.html#R4" title="RELAT_2:pred.4">is_antisymmetric_in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">L</font>
 <b>by </b><i><a class="txt" href="orders_2.html#E4:4_1_1"><i><font color="Green" title="E5">A1</font></i></a>, <a class="ref" href="relat_2.html#D12" title="RELAT_2:def.12">RELAT_2:def 12</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="orders_2.html#D6" target="_self" title="ORDERS_2:def.6">ORDERS_2:def 6</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">L</font> is <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a> &amp; <font color="Maroon" title="c3">L</font> is <a href="orders_2.html#V2" title="ORDERS_2:attr.2">total</a> )</span><br/>

<b>thus </b><a NAME="E8:4_1_1"/>
<font color="Maroon" title="c3">L</font> is <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">L</font> is <a href="orders_2.html#V2" title="ORDERS_2:attr.2">total</a></span><br/>

<b>thus </b><a NAME="E9:4_1_1"/>
the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c3">L</font> is <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="orders_2.html#D3" target="_self" title="ORDERS_2:def.3">ORDERS_2:def 3</a> <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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