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<b>let </b><font color="Maroon">A</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</font> be   <a href="orders_2.html#NM2">Chain</a> of <font color="Maroon">A</font>;<br/>



<b>set </b><font color="Maroon">P</font> = the <a href="orders_2.html#U1" target="_self">InternalRel</a> of <font color="Maroon">A</font> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">C</font>;<br/>
<a NAME="E1:72"/>
the <a href="orders_2.html#U1" target="_self">InternalRel</a> of <font color="Maroon">A</font> <a href="orders_1.html#NR2" target="_self">is_partial-order</a> 
 
<b>by </b><i><a class="ref" href="orders_1.html#D4">ORDERS_1:def 4</a></i>;<br/>
<a NAME="E2:72"/><b>then </b>
the <a href="orders_2.html#U1" target="_self">InternalRel</a> of <font color="Maroon">A</font> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">C</font> <a href="orders_1.html#NR2" target="_self">is_partial-order</a> 
 
<b>by </b><i><a class="ref" href="orders_1.html#T114">ORDERS_1:114</a></i>;<br/>
<b>hence </b><a NAME="E3:72"/>
( the <a href="orders_2.html#U1" target="_self">InternalRel</a> of <font color="Maroon">A</font> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">C</font> is <a href="relat_2.html#V1">reflexive</a> &amp; the <a href="orders_2.html#U1" target="_self">InternalRel</a> of <font color="Maroon">A</font> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">C</font> is <a href="relat_2.html#V8">transitive</a> &amp; the <a href="orders_2.html#U1" target="_self">InternalRel</a> of <font color="Maroon">A</font> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">C</font> is <a href="relat_2.html#V4">antisymmetric</a> )
 <b>by </b><i><a class="ref" href="orders_1.html#D4">ORDERS_1:def 4</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="orders_1.html#D5">ORDERS_1:def 5</a><br/>

<a NAME="E4:72"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1" target="_self">InternalRel</a> of <font color="Maroon">A</font> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">C</font></span>)</span>
 
<b>by </b><i><a class="ref" href="orders_2.html#T35" target="_self">Th35</a></i>;<br/>
<a NAME="E5:72"/>
the <a href="orders_2.html#U1" target="_self">InternalRel</a> of <font color="Maroon">A</font> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">C</font> <a href="relat_2.html#R6">is_connected_in</a> <font color="Maroon">C</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:72_1"/><i><font color="Green">E19</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> )
 ;<br/>

<a NAME="E2:72_1"/>
the <a href="orders_2.html#U1" target="_self">InternalRel</a> of <font color="Maroon">A</font> <a href="relat_2.html#R7">is_strongly_connected_in</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="orders_2.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E3:72_1"/><b>then </b><i><font color="Green">E39</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1" target="_self">InternalRel</a> of <font color="Maroon">A</font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1" target="_self">InternalRel</a> of <font color="Maroon">A</font> )
 
<b>by </b><i>, <a class="ref" href="relat_2.html#D7">RELAT_2:def 7</a></i>;<br/>
<a NAME="E4:72_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">C</font>,<font color="Maroon">C</font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">C</font>,<font color="Maroon">C</font></span><a href="zfmisc_1.html#K2">:]</a></span> )
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>hence </b><a NAME="E5:72_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1" target="_self">InternalRel</a> of <font color="Maroon">A</font> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">C</font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1" target="_self">InternalRel</a> of <font color="Maroon">A</font> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">C</font> )
 <b>by </b><i><a class="ref" href="orders_2.html#T38" target="_self">Th38</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:72"/>
the <a href="orders_2.html#U1" target="_self">InternalRel</a> of <font color="Maroon">A</font> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">C</font> is <a href="relat_2.html#V6">connected</a>
 <b>by </b><i><a class="ref" href="orders_2.html#T36" target="_self">Th36</a>, <a class="ref" href="relat_2.html#D14">RELAT_2:def 14</a></i>;<br/>


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