<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">n</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="mycielsk.html#K3" title="MYCIELSK:func.3">cliquecover#</a> <span class="p1">(<span class="default"><a href="mycielsk.html#K5" title="MYCIELSK:func.5">CompleteRelStr</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/>

<span class="kw">set </span><font color="Maroon" title="c2">R</font> =  <a href="mycielsk.html#K5" title="MYCIELSK:func.5">CompleteRelStr</a> <font color="Maroon" title="c1">n</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c3">cR</font> =  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="mycielsk.html#K5" title="MYCIELSK:func.5">CompleteRelStr</a> <font color="Maroon" title="c1">n</font></span>)</span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c4">C</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="mycielsk.html#K5" title="MYCIELSK:func.5">CompleteRelStr</a> <font color="Maroon" title="c1">n</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> as    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="mycielsk.html#K5" title="MYCIELSK:func.5">CompleteRelStr</a> <font color="Maroon" title="c1">n</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="eqrel_1.html#T39" title="EQREL_1:th.39">EQREL_1:39</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E36">A1</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> implies <font color="Maroon" title="c5">x</font> is   <a href="dilworth.html#NM1" title="DILWORTH:NM.1">Clique</a> of <span class="p1">(<span class="default"><a href="mycielsk.html#K5" title="MYCIELSK:func.5">CompleteRelStr</a> <font color="Maroon" title="c1">n</font></span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E1:68_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">x</font> is   <a href="dilworth.html#NM1" title="DILWORTH:NM.1">Clique</a> of <span class="p1">(<span class="default"><a href="mycielsk.html#K5" title="MYCIELSK:func.5">CompleteRelStr</a> <font color="Maroon" title="c1">n</font></span>)</span></span><br/><a NAME="E2:68_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="mycielsk.html#K5" title="MYCIELSK:func.5">CompleteRelStr</a> <font color="Maroon" title="c1">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E3:68_1"/>
<font color="Maroon" title="c5">x</font> is   <a href="dilworth.html#NM1" title="DILWORTH:NM.1">Clique</a> of <span class="p1">(<span class="default"><a href="mycielsk.html#K5" title="MYCIELSK:func.5">CompleteRelStr</a> <font color="Maroon" title="c1">n</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E37">A2</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="68_2"><span class="kw">now </span></a><div class="add"><span class="kw">take </span><font color="Maroon" title="c5">C</font> = <font color="Maroon" title="c4">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">C</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  &amp; <font color="Maroon" title="c5">C</font> is   <a href="dilworth.html#NM3" title="DILWORTH:NM.3">Clique-partition</a> of <span class="p1">(<span class="default"><a href="mycielsk.html#K5" title="MYCIELSK:func.5">CompleteRelStr</a> <font color="Maroon" title="c1">n</font></span>)</span> &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c5">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/><span class="kw">thus </span><a NAME="E1:68_2"/>
<font color="Maroon" title="c5">C</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">C</font> is   <a href="dilworth.html#NM3" title="DILWORTH:NM.3">Clique-partition</a> of <span class="p1">(<span class="default"><a href="mycielsk.html#K5" title="MYCIELSK:func.5">CompleteRelStr</a> <font color="Maroon" title="c1">n</font></span>)</span> &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c5">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/><span class="kw">thus </span><a NAME="E2:68_2"/>
<font color="Maroon" title="c5">C</font> is   <a href="dilworth.html#NM3" title="DILWORTH:NM.3">Clique-partition</a> of <span class="p1">(<span class="default"><a href="mycielsk.html#K5" title="MYCIELSK:func.5">CompleteRelStr</a> <font color="Maroon" title="c1">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:68"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="ref" href="dilworth.html#D11" title="DILWORTH:def.11">DILWORTH:def 11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c5">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><span class="kw">thus </span><a NAME="E3:68_2"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c5">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T30" title="CARD_1:th.30">CARD_1:30</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">C</font> be   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="dilworth.html#NM3" title="DILWORTH:NM.3">Clique-partition</a> of <span class="p1">(<span class="default"><a href="mycielsk.html#K5" title="MYCIELSK:func.5">CompleteRelStr</a> <font color="Maroon" title="c1">n</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c5">C</font></span><br/><a NAME="E1:68_3"/>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c5">C</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><span class="kw">hence </span><a NAME="E2:68_3"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c5">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:68"/>
 <a href="mycielsk.html#K3" title="MYCIELSK:func.3">cliquecover#</a> <span class="p1">(<span class="default"><a href="mycielsk.html#K5" title="MYCIELSK:func.5">CompleteRelStr</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:68"><span class="lab"><font color="Green" title="E37">A2</font></span></a>, <a class="ref" href="mycielsk.html#D5" target="_self" title="MYCIELSK:def.5">Def5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
