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<b>let </b><font color="Maroon" title="c1">A</font> be  <a href="orders_2.html#V2" title="ORDERS_2:attr.2">reflexive</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ;<br/><b>let </b><font color="Maroon" title="c2">a1</font>, <font color="Maroon" title="c3">a2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font>;<br/>



<b>thus </b><a NAME="E1:27"/>
( for <font color="Olive" title="b1">C</font> being  <a href="orders_2.html#NM2" title="ORDERS_2:NM.2">Chain</a> of <font color="Maroon" title="c1">A</font>  holds <br/>(  not <font color="Maroon" title="c2">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> or  not <font color="Maroon" title="c3">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> ) or <font color="Maroon" title="c2">a1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c3">a2</font> or <font color="Maroon" title="c3">a2</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">a1</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon" title="c4">C</font> being   <a href="orders_2.html#NM2" title="ORDERS_2:NM.2">Chain</a> of <font color="Maroon" title="c1">A</font><b> such that </b><a NAME="E1:27_1"/><i><font color="Green" title="E18">A1</font></i>: 
( <font color="Maroon" title="c2">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Maroon" title="c3">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> )
 ;<br/>

<a NAME="E2:27_1"/>
the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">A</font> <a href="relat_2.html#R7" title="RELAT_2:pred.7">is_strongly_connected_in</a> <font color="Maroon" title="c4">C</font>
 
<b>by </b><i><a class="ref" href="orders_2.html#D11" target="_self" title="ORDERS_2:def.11">Def11</a></i>;<br/>
<a NAME="E3:27_1"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c3">a2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">A</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c2">a1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">A</font> )
 
<b>by </b><i><a class="txt" href="orders_2.html#E1:27_1"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="relat_2.html#D7" title="RELAT_2:def.7">RELAT_2:def 7</a></i>;<br/>
<b>hence </b><a NAME="E4:27_1"/>
( <font color="Maroon" title="c2">a1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c3">a2</font> or <font color="Maroon" title="c3">a2</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">a1</font> )
 <b>by </b><i><a class="ref" href="orders_2.html#D9" target="_self" title="ORDERS_2:def.9">Def9</a></i>;<br/>


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

<b>assume </b><a NAME="E2:27"/><i><font color="Green" title="E18">A2</font></i>: 
( <font color="Maroon" title="c2">a1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c3">a2</font> or <font color="Maroon" title="c3">a2</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">a1</font> )
 ;<br/>

<a NAME="E3:27"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c3">a2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">A</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c2">a1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">A</font> )
 
<b>by </b><i><a class="ref" href="orders_2.html#D9" target="_self" title="ORDERS_2:def.9">Def9</a></i>;<br/>
<a NAME="E4:27"/><b>then </b>
<font color="Maroon" title="c2">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c4">B</font> = <font color="Maroon" title="c1">A</font> as  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="orders_2.html#V2" title="ORDERS_2:attr.2">reflexive</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  <b>by </b><i><a class="ref" href="struct_0.html#D1" title="STRUCT_0:def.1">STRUCT_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">b1</font> = <font color="Maroon" title="c2">a1</font>, <font color="Maroon" title="c6">b2</font> = <font color="Maroon" title="c3">a2</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">B</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c7">Y</font> = <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c5">b1</font>,<font color="Maroon" title="c6">b2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> as   <a href="orders_2.html#NM2" title="ORDERS_2:NM.2">Chain</a> of <font color="Maroon" title="c1">A</font> <b>by </b><i><a class="txt" href="orders_2.html#E2:27"><i><font color="Green" title="E18">A2</font></i></a>, <a class="ref" href="orders_2.html#T36" target="_self" title="ORDERS_2:th.36">Th36</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c7">Y</font>
;<br/>

<b>thus </b><a NAME="E8:27"/>
( <font color="Maroon" title="c2">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Y</font> &amp; <font color="Maroon" title="c3">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Y</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>


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