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<span class="kw">consider </span><font color="Maroon" title="c2">X</font> being   <a href="lattice3.html#V1" title="LATTICE3:attr.1">with_suprema</a>  <a href="orders_2.html#NM1" title="ORDERS_2:NM.1">Poset</a>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c3">M</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>;<br/>
<span class="kw">take </span><font color="Maroon" title="c4">N</font> =  <a href="waybel_0.html#G1" title="WAYBEL_0:aggr.1">NetStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c3">M</font> #); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( not <font color="Maroon" title="c4">N</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  &amp; <font color="Maroon" title="c4">N</font> is  <a href="orders_2.html#V3" title="ORDERS_2:attr.3">reflexive</a>  &amp; <font color="Maroon" title="c4">N</font> is  <a href="orders_2.html#V4" title="ORDERS_2:attr.4">transitive</a>  &amp; <font color="Maroon" title="c4">N</font> is  <a href="orders_2.html#V5" title="ORDERS_2:attr.5">antisymmetric</a>  &amp; <font color="Maroon" title="c4">N</font> is  <a href="waybel_0.html#V7" title="WAYBEL_0:attr.7">directed</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E3:23_1_1"/>
not <font color="Maroon" title="c4">N</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">N</font> is  <a href="orders_2.html#V3" title="ORDERS_2:attr.3">reflexive</a>  &amp; <font color="Maroon" title="c4">N</font> is  <a href="orders_2.html#V4" title="ORDERS_2:attr.4">transitive</a>  &amp; <font color="Maroon" title="c4">N</font> is  <a href="orders_2.html#V5" title="ORDERS_2:attr.5">antisymmetric</a>  &amp; <font color="Maroon" title="c4">N</font> is  <a href="waybel_0.html#V7" title="WAYBEL_0:attr.7">directed</a>  )</span><br/>

<a NAME="E4:23_1_1"/><span class="lab"><font color="Green" title="E10">A1</font></span>: 
the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c4">N</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="c4">N</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="orders_2.html#D4" title="ORDERS_2:def.4">ORDERS_2:def 4</a></span>;<br/>
<a NAME="E5:23_1_1"/><span class="lab"><font color="Green" title="E11">A2</font></span>: 
the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c4">N</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="c4">N</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="orders_2.html#D5" title="ORDERS_2:def.5">ORDERS_2:def 5</a></span>;<br/>
<a NAME="E6:23_1_1"/>
the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c4">N</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="c4">N</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="orders_2.html#D6" title="ORDERS_2:def.6">ORDERS_2:def 6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:23_1_1"/>
( <font color="Maroon" title="c4">N</font> is  <a href="orders_2.html#V3" title="ORDERS_2:attr.3">reflexive</a>  &amp; <font color="Maroon" title="c4">N</font> is  <a href="orders_2.html#V4" title="ORDERS_2:attr.4">transitive</a>  &amp; <font color="Maroon" title="c4">N</font> is  <a href="orders_2.html#V5" title="ORDERS_2:attr.5">antisymmetric</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:23_1_1"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="txt" href="#E5:23_1_1"><span class="lab"><font color="Green" title="E11">A2</font></span></a>, <a class="ref" href="orders_2.html#D4" title="ORDERS_2:def.4">ORDERS_2:def 4</a>, <a class="ref" href="orders_2.html#D5" title="ORDERS_2:def.5">ORDERS_2:def 5</a>, <a class="ref" href="orders_2.html#D6" title="ORDERS_2:def.6">ORDERS_2:def 6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">N</font> is  <a href="waybel_0.html#V7" title="WAYBEL_0:attr.7">directed</a> </span><br/>

<a NAME="E8:23_1_1"/><span class="lab"><font color="Green" title="E12">A3</font></span>: 
 <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="c4">N</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c4">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 <font color="Maroon" title="c2">X</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c2">X</font> #)
 
;<br/>
<a NAME="E9:23_1_1"/>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c4">N</font>
 
;<br/>
<span class="kw">hence </span><a NAME="E10:23_1_1"/>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c4">N</font> is  <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:23_1_1"><span class="lab"><font color="Green" title="E12">A3</font></span></a>, <a class="ref" href="waybel_0.html#T3" target="_self" title="WAYBEL_0:th.3">Th3</a></span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="waybel_0.html#D6" target="_self" title="WAYBEL_0:def.6">WAYBEL_0:def 6</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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