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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#NM1">Poset</a>;<br/>

<b>set </b><font color="Maroon">c<sub>2</sub></font> = the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>3</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<a NAME="E1:38"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="dickson.html#V3" target="_self">quasi_ordered</a>
 
<b>by </b><i><a class="ref" href="dickson.html#D3" target="_self">Def3</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:38_1"/>
the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="wellord1.html#R2">well_orders</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/><a NAME="E2:38_1"/><b>then </b><i><font color="Green">E24</font></i>: 
( the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="relat_2.html#R1">is_reflexive_in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp; the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="relat_2.html#R8">is_transitive_in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp; the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="relat_2.html#R4">is_antisymmetric_in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp; the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="relat_2.html#R6">is_connected_in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp; the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="wellord1.html#R1">is_well_founded_in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="ref" href="wellord1.html#D5">WELLORD1:def 5</a></i>;<br/><a NAME="E3:38_1"/><b>then </b>
the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="relat_2.html#R7">is_strongly_connected_in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="orders_1.html#T92">ORDERS_1:92</a></i>;<br/><a NAME="E4:38_1"/><b>then </b><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="waybel_0.html#V16">connected</a>
 <b>by </b><i><a class="ref" href="dickson.html#T5" target="_self">Th5</a></i>;<br/><a NAME="E5:38_1"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="wellfnd1.html#V1">well_founded</a>
 <b>by </b><i><a class="txt" href="dickson.html#E2:38_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="wellfnd1.html#D2">WELLFND1:def 2</a></i>;<br/><a NAME="E6:38_1"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="dickson.html#K5" target="_self">\~</a>  is <a href="wellfnd1.html#V1">well_founded</a>
 <b>by </b><i><a class="ref" href="dickson.html#T16" target="_self">Th16</a></i>;<br/><b>hence </b><a NAME="E7:38_1"/>
for <font color="Olive">b<sub>1</sub></font> being non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> holds  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="dickson.html#K6" target="_self">min-classes</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="dickson.html#E1:38"><i><font color="Green">E23</font></i></a>, <a class="txt" href="dickson.html#E4:38_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="dickson.html#T24" target="_self">Th24</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E3:38"/>
for <font color="Olive">b<sub>1</sub></font> being non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> holds  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="dickson.html#K6" target="_self">min-classes</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 1
 ;<br/>

<a NAME="E4:38"/><b>then </b><i><font color="Green">E24</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> is <a href="waybel_0.html#V16">connected</a> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="dickson.html#K5" target="_self">\~</a>  is <a href="wellfnd1.html#V1">well_founded</a> )
 
<b>by </b><i><a class="txt" href="dickson.html#E1:38"><i><font color="Green">E23</font></i></a>, <a class="ref" href="dickson.html#T24" target="_self">Th24</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_2"><b>now </b></a><div class="add"><b>thus </b><a NAME="E1:38_2"/>
the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="relat_2.html#R1">is_reflexive_in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="orders_2.html#D4">ORDERS_2:def 4</a></i>;<br/><b>thus </b><a NAME="E2:38_2"/>
the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="relat_2.html#R8">is_transitive_in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="orders_2.html#D5">ORDERS_2:def 5</a></i>;<br/><b>thus </b><a NAME="E3:38_2"/>
the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="relat_2.html#R4">is_antisymmetric_in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="orders_2.html#D6">ORDERS_2:def 6</a></i>;<br/><a NAME="E4:38_2"/>
the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="relat_2.html#R7">is_strongly_connected_in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E4:38"><i><font color="Green">E24</font></i></a>, <a class="ref" href="dickson.html#T5" target="_self">Th5</a></i>;<br/><b>hence </b><a NAME="E5:38_2"/>
the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="relat_2.html#R6">is_connected_in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="orders_1.html#T92">ORDERS_1:92</a></i>;<br/><a NAME="E6:38_2"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="wellfnd1.html#V1">well_founded</a>
 <b>by </b><i><a class="txt" href="dickson.html#E4:38"><i><font color="Green">E24</font></i></a>, <a class="ref" href="dickson.html#T17" target="_self">Th17</a></i>;<br/><b>hence </b><a NAME="E7:38_2"/>
the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="wellord1.html#R1">is_well_founded_in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="wellfnd1.html#D2">WELLFND1:def 2</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:38"/>
the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="wellord1.html#R2">well_orders</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="wellord1.html#D5">WELLORD1:def 5</a></i>;<br/>


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