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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>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E1:15"/><i><font color="Green">E7</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="osalg_4.html#R1" target="_self">~=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="osalg_4.html#R1" target="_self">~=</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/>

<b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="osalg_4.html#K1" target="_self">Path_Rel</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<a NAME="E2:15"/><i><font color="Green">E8</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>2</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="osalg_4.html#K1" target="_self">Path_Rel</a> <font color="Maroon">c<sub>1</sub></font> &amp; <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="osalg_4.html#K1" target="_self">Path_Rel</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="osalg_4.html#E1:15"><i><font color="Green">E7</font></i></a>, <a class="ref" href="osalg_4.html#D5" target="_self">Def5</a></i>;<br/>
<a NAME="E3:15"/>
 <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="osalg_4.html#K1" target="_self">Path_Rel</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</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#T97">ORDERS_1:97</a></i>;<br/>
<a NAME="E4:15"/><b>then </b>
 <a href="osalg_4.html#K1" target="_self">Path_Rel</a> <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="relat_2.html#D16">RELAT_2:def 16</a></i>;<br/>
<a NAME="E5:15"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>2</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="osalg_4.html#K1" target="_self">Path_Rel</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="osalg_4.html#E2:15"><i><font color="Green">E8</font></i></a>, <a class="ref" href="relat_2.html#D8">RELAT_2:def 8</a></i>;<br/>
<b>hence </b><a NAME="E6:15"/>
<font color="Maroon">c<sub>2</sub></font> <a href="osalg_4.html#R1" target="_self">~=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="osalg_4.html#D5" target="_self">Def5</a></i>;<br/>


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