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<b>let </b><font color="Maroon" title="c1">R</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ;<br/>

<b>set </b><font color="Maroon" title="c2">IR</font> = the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font>;<br/>
<b>set </b><font color="Maroon" title="c3">CR</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:8_1"/><i><font color="Green" title="E7">A1</font></i>: 
<font color="Maroon" title="c1">R</font> is <a href="waybel_0.html#V16" title="WAYBEL_0:attr.16">connected</a>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:8_1_1"/><i><font color="Green" title="E8">A2</font></i>: 
( <font color="Maroon" title="c4">x</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">R</font> &amp; <font color="Maroon" title="c5">y</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">R</font> )
 ;<br/><b>reconsider </b><font color="Maroon" title="c6">x'</font> = <font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c7">y'</font> = <font color="Maroon" title="c5">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> <b>by </b><i><a class="txt" href="dickson.html#E1:8_1_1"><i><font color="Green" title="E8">A2</font></i></a></i>;<br/><a NAME="E3:8_1_1"/>
( <font color="Maroon" title="c6">x'</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c7">y'</font> or <font color="Maroon" title="c7">y'</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c6">x'</font> )
 <b>by </b><i><a class="txt" href="dickson.html#E1:8_1"><i><font color="Green" title="E7">A1</font></i></a>, <a class="ref" href="waybel_0.html#D29" title="WAYBEL_0:def.29">WAYBEL_0:def 29</a></i>;<br/><b>hence </b><a NAME="E4:8_1_1"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</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">R</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">x</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">R</font> )
 <b>by </b><i><a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:8_1"/>
the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="relat_2.html#R7" title="RELAT_2:pred.7">is_strongly_connected_in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="ref" href="relat_2.html#D7" title="RELAT_2:def.7">RELAT_2:def 7</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:8"/><i><font color="Green" title="E7">A3</font></i>: 
the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="relat_2.html#R7" title="RELAT_2:pred.7">is_strongly_connected_in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>;<br/><a NAME="E1:8_2"/>
( <span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</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">R</font> or <span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">x</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</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">R</font> )
 <b>by </b><i><a class="txt" href="dickson.html#E2:8"><i><font color="Green" title="E7">A3</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="E2:8_2"/>
( <font color="Maroon" title="c4">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">y</font> or <font color="Maroon" title="c5">y</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c4">x</font> )
 <b>by </b><i><a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:8"/>
<font color="Maroon" title="c1">R</font> is <a href="waybel_0.html#V16" title="WAYBEL_0:attr.16">connected</a>
 <b>by </b><i><a class="ref" href="waybel_0.html#D29" title="WAYBEL_0:def.29">WAYBEL_0:def 29</a></i>;<br/>


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