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<span class="kw">let </span><font color="Maroon" title="c2">C</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">R</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">C</font> is  <a href="dilworth.html#V7" title="DILWORTH:attr.7">strong-chain</a>  implies <font color="Maroon" title="c2">C</font> is  <a href="dilworth.html#V1" title="DILWORTH:attr.1">connected</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:98_1_1"/><span class="lab"><font color="Green" title="E55">A</font></span>: 
<font color="Maroon" title="c2">C</font> is  <a href="dilworth.html#V7" title="DILWORTH:attr.7">strong-chain</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">C</font> is  <a href="dilworth.html#V1" title="DILWORTH:attr.1">connected</a> </span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font> &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font> &amp; <font color="Maroon" title="c3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> &amp; not <font color="Maroon" title="c3">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c4">b</font> implies <font color="Maroon" title="c4">b</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c3">a</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:98_1_1_1"/><span class="lab"><font color="Green" title="E56">A1</font></span>: 
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font>
 <span class="kw">and </span><br/><a NAME="E2:98_1_1_1"/><span class="lab"><font color="Green" title="E57">B1</font></span>: 
<font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font>
 <span class="kw">and </span><br/><a NAME="E3:98_1_1_1"/><span class="lab"><font color="Green" title="E58">B1a</font></span>: 
<font color="Maroon" title="c3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c4">b</font> or <font color="Maroon" title="c4">b</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c3">a</font> )</span><br/><span class="kw">set </span><font color="Maroon" title="c5">S</font> = <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c6">S</font> = <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> as  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">R</font> ;<br/><span class="kw">consider </span><font color="Maroon" title="c7">P</font> being   <a href="dilworth.html#NM3" title="DILWORTH:NM.3">Clique-partition</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c6">S</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E6:98_1_1_1"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c7">P</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="dilworth.html#K2" title="DILWORTH:func.2">stability#</a> <font color="Maroon" title="c1">R</font>
 <span class="kw">and </span><br/><a NAME="E7:98_1_1_1"/><span class="lab"><font color="Green" title="E59">C1</font></span>: 
 ex <font color="Olive" title="b1">c</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">P</font> &amp; <font color="Maroon" title="c6">S</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">c</font> &amp; ( for <font color="Olive" title="b2">d</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">P</font> &amp; <font color="Olive" title="b2">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">c</font> holds <br/><font color="Maroon" title="c2">C</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Olive" title="b2">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:98_1_1"><span class="lab"><font color="Green" title="E55">A</font></span></a>, <a class="ref" href="dilworth.html#D13" target="_self" title="DILWORTH:def.13">Lsc</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c8">c</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E9:98_1_1_1"/><span class="lab"><font color="Green" title="E60">D1</font></span>: 
<font color="Maroon" title="c8">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">P</font>
 <span class="kw">and </span><br/><a NAME="E10:98_1_1_1"/><span class="lab"><font color="Green" title="E61">E1</font></span>: 
<font color="Maroon" title="c6">S</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">c</font>
 <span class="kw">and </span><br/><a NAME="E11:98_1_1_1"/>
for <font color="Olive" title="b1">d</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">P</font> &amp; <font color="Olive" title="b1">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">c</font> holds <br/><font color="Maroon" title="c2">C</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Olive" title="b1">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:98_1_1_1"><span class="lab"><font color="Green" title="E59">C1</font></span></a></span>;<br/><a NAME="E12:98_1_1_1"/>
<font color="Maroon" title="c8">c</font> is   <a href="dilworth.html#NM1" title="DILWORTH:NM.1">Clique</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c6">S</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:98_1_1_1"><span class="lab"><font color="Green" title="E60">D1</font></span></a>, <a class="ref" href="dilworth.html#D11" target="_self" title="DILWORTH:def.11">LCpart</a></span>;<br/><a NAME="E13:98_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">F1</font></span>: 
<font color="Maroon" title="c8">c</font> is   <a href="dilworth.html#NM1" title="DILWORTH:NM.1">Clique</a> of <font color="Maroon" title="c1">R</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="dilworth.html#T28" target="_self" title="DILWORTH:th.28">SPClique</a></span>;<br/><a NAME="E14:98_1_1_1"/>
( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">S</font> &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">S</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/><a NAME="E15:98_1_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">S</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">C</font> &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">S</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">C</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:98_1_1_1"><span class="lab"><font color="Green" title="E56">A1</font></span></a>, <a class="txt" href="#E2:98_1_1_1"><span class="lab"><font color="Green" title="E57">B1</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E16:98_1_1_1"/>
( <font color="Maroon" title="c3">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c4">b</font> or <font color="Maroon" title="c4">b</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c3">a</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:98_1_1_1"><span class="lab"><font color="Green" title="E61">E1</font></span></a>, <a class="txt" href="#E13:98_1_1_1"><span class="lab"><font color="Green" title="E62">F1</font></span></a>, <a class="txt" href="#E3:98_1_1_1"><span class="lab"><font color="Green" title="E58">B1a</font></span></a>, <a class="ref" href="dilworth.html#T6" target="_self" title="DILWORTH:th.6">DClique</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:98_1_1"/>
<font color="Maroon" title="c2">C</font> is  <a href="dilworth.html#V1" title="DILWORTH:attr.1">connected</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="dilworth.html#T6" target="_self" title="DILWORTH:th.6">DClique</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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