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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">f</font> be    <a href="orders_1.html#M1">Choice_Function</a> of  <a href="orders_1.html#K1">BOOL</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>;<br/>



<b>reconsider </b><font color="Maroon">C</font> =  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="orders_2.html#K4" target="_self">Chains</a> <font color="Maroon">f</font></span>)</span> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">A</font> <b>by </b><i/>;<br/>
<a NAME="E2:66"/>
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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_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#D7">RELAT_2:def 7</a><br/><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:66_1"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 <b>and </b><br/><a NAME="E2:66_1"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 ;<br/>

<b>consider </b><font color="Maroon">X</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:66_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E5:66_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="orders_2.html#K4" target="_self">Chains</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="orders_2.html#D11" target="_self">Def11</a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">Y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:66_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>and </b><br/><a NAME="E8:66_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a>  <a href="orders_2.html#K4" target="_self">Chains</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">X</font> = <font color="Maroon">X</font>, <font color="Maroon">Y</font> = <font color="Maroon">Y</font> as    <a href="orders_2.html#M2" target="_self">Chain</a> of <font color="Maroon">f</font> <b>by </b><i><a class="ref" href="orders_2.html#T35" target="_self">Th35</a>, , <a class="txt" href="orders_2.html#E10:66_1"><i><font color="Green">E43</font></i></a></i>;<br/>
<a NAME="E10:66_1"/><i><font color="Green">E43</font></i>: 
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">X</font>
 
<b>by </b><i><a class="ref" href="orders_2.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E11:66_1"/><i><font color="Green">E44</font></i>: 
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">Y</font>
 
<b>by </b><i><a class="ref" href="orders_2.html#D4" target="_self">Def4</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:66_1_1_1"/>
( <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font> or <font color="Maroon">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">Y</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_1_1_1"><b>suppose </b></a><a NAME="E1:66_1_1_1_1"/>
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:66_1_1_1_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> 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="orders_2.html#T36" target="_self">Th36</a>, <a class="ref" href="orders_2.html#T38" target="_self">Th38</a>, <a class="ref" href="relat_2.html#D7">RELAT_2:def 7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_1_1_2"><b>suppose </b></a><a NAME="E1:66_1_1_1_2"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">Y</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:66_1_1_1_2_1_1"/>
( <font color="Maroon">X</font> is    <a href="orders_2.html#M1" target="_self">Initial_Segm</a> of <font color="Maroon">Y</font> or <font color="Maroon">Y</font> is    <a href="orders_2.html#M1" target="_self">Initial_Segm</a> of <font color="Maroon">X</font> )
 </span><b>by </b><i>, <a class="txt" href="orders_2.html#E7:66_1"><i><font color="Green">E41</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:66_1_1_1_2_1_1_1"/>
<font color="Maroon">X</font> is    <a href="orders_2.html#M1" target="_self">Initial_Segm</a> of <font color="Maroon">Y</font>
 ;<br/><div class="add"><a NAME="E2:66_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">X</font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="txt" href="orders_2.html#E8:66_1"><i><font color="Green">E42</font></i></a></i>;<br/><a NAME="E3:66_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/><b>hence </b><a NAME="E4:66_1_1_1_2_1_1_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> 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="orders_2.html#T36" target="_self">Th36</a>, , <a class="ref" href="relat_2.html#D7">RELAT_2:def 7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:66_1_1_1_2_1_1_2"/>
<font color="Maroon">Y</font> is    <a href="orders_2.html#M1" target="_self">Initial_Segm</a> of <font color="Maroon">X</font>
 ;<br/><div class="add"><a NAME="E2:66_1_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">Y</font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="txt" href="orders_2.html#E8:66_1"><i><font color="Green">E42</font></i></a></i>;<br/><a NAME="E3:66_1_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/><b>hence </b><a NAME="E4:66_1_1_1_2_1_1_2"/>
( <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="orders_2.html#T36" target="_self">Th36</a>, <a class="ref" href="orders_2.html#T38" target="_self">Th38</a>, <a class="ref" href="relat_2.html#D7">RELAT_2:def 7</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:66_1_1_1_2"/>
( <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> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E13:66_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> 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> )
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:66"/>
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="orders_2.html#K4" target="_self">Chains</a> <font color="Maroon">f</font></span>)</span> is   <a href="orders_2.html#NM2">Chain</a> of <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="orders_2.html#D4" target="_self">Def4</a></i>;<br/>


</div>
