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<div class="add">

<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#L1">RelStr</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:16"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="dickson.html#V3" target="_self">quasi_ordered</a>
 <b>and </b><br/><a NAME="E2:16"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="waybel_0.html#V16">connected</a>
 ;<br/>

<a NAME="E3:16"/><i><font color="Green">E16</font></i>: 
 <a href="dickson.html#K2" target="_self">&lt;=E</a> <font color="Maroon">c<sub>1</sub></font> <a href="orders_1.html#R2">partially_orders</a>  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="dickson.html#K1" target="_self">EqRel</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="dickson.html#E1:16"><i><font color="Green">E14</font></i></a>, <a class="ref" href="dickson.html#T10" target="_self">Th10</a></i>;<br/>
<b>hence </b><a NAME="E4:16"/>
 <a href="dickson.html#K2" target="_self">&lt;=E</a> <font color="Maroon">c<sub>1</sub></font> <a href="relat_2.html#R1">is_reflexive_in</a>  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="dickson.html#K1" target="_self">EqRel</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="orders_1.html#D7">ORDERS_1:def 7</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="orders_1.html#D8">ORDERS_1:def 8</a><br/>

<b>thus </b><a NAME="E5:16"/>
 <a href="dickson.html#K2" target="_self">&lt;=E</a> <font color="Maroon">c<sub>1</sub></font> <a href="relat_2.html#R8">is_transitive_in</a>  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="dickson.html#K1" target="_self">EqRel</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E3:16"><i><font color="Green">E16</font></i></a>, <a class="ref" href="orders_1.html#D7">ORDERS_1:def 7</a></i>;<br/>

<b>thus </b><a NAME="E6:16"/>
 <a href="dickson.html#K2" target="_self">&lt;=E</a> <font color="Maroon">c<sub>1</sub></font> <a href="relat_2.html#R4">is_antisymmetric_in</a>  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="dickson.html#K1" target="_self">EqRel</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E3:16"><i><font color="Green">E16</font></i></a>, <a class="ref" href="orders_1.html#D7">ORDERS_1:def 7</a></i>;<br/>

<b>thus </b><a NAME="E7:16"/>
 <a href="dickson.html#K2" target="_self">&lt;=E</a> <font color="Maroon">c<sub>1</sub></font> <a href="relat_2.html#R6">is_connected_in</a>  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="dickson.html#K1" target="_self">EqRel</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">c<sub>2</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="relat_2.html#D6">RELAT_2:def 6</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:16_1"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="dickson.html#K1" target="_self">EqRel</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E2:16_1"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="dickson.html#K1" target="_self">EqRel</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E3:16_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E4:16_1"/><i><font color="Green">E19</font></i>: 
not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="dickson.html#K2" target="_self">&lt;=E</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:16_1"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E7:16_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="dickson.html#K1" target="_self">EqRel</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E1:16_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="eqrel_1.html#D5">EQREL_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E9:16_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E10:16_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="dickson.html#K1" target="_self">EqRel</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E2:16_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="eqrel_1.html#D5">EQREL_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="dickson.html#E6:16_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="dickson.html#E9:16_1"><i><font color="Green">E22</font></i></a></i>;<br/>
<a NAME="E12:16_1"/>
not <font color="Maroon">c<sub>7</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="dickson.html#E4:16_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="dickson.html#E7:16_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="dickson.html#E10:16_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="dickson.html#D5" target="_self">Def5</a></i>;<br/>
<a NAME="E13:16_1"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="dickson.html#E2:16"><i><font color="Green">E15</font></i></a>, <a class="ref" href="waybel_0.html#D29">WAYBEL_0:def 29</a></i>;<br/>
<b>hence </b><a NAME="E14:16_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="dickson.html#K2" target="_self">&lt;=E</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E7:16_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="dickson.html#E10:16_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="dickson.html#D5" target="_self">Def5</a></i>;<br/>


</div><b>end;</b></div>


</div>
