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<span class="kw">consider </span><font color="Maroon" title="c3">C</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="dilworth.html#NM1" title="DILWORTH:NM.1">Clique</a> of <font color="Maroon" title="c1">R</font><span class="kw"> such that </span><br/><a NAME="E2:71_1_1"/><span class="lab"><font color="Green" title="E40">A</font></span>: 
for <font color="Olive" title="b1">D</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="dilworth.html#NM1" title="DILWORTH:NM.1">Clique</a> of <font color="Maroon" title="c1">R</font> holds   <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">D</font> <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="c3">C</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="dilworth.html#D3" target="_self" title="DILWORTH:def.3">Lfheight</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">sS</font> =  <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">S</font>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span> ex <font color="Olive" title="b1">c</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <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="c2">S</font></span>)</span> st  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">R</font>;<br/>
<a NAME="E3:71_1_1"/><span class="lab"><font color="Green" title="E41">P1</font></span>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Olive" title="b1">k</font> <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="c3">C</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">k</font> be   <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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font>] implies <font color="Maroon" title="c5">k</font> <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="c3">C</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:71_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</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="c5">k</font> <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="c3">C</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">c</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <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="c2">S</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E3:71_1_1_1"/><span class="lab"><font color="Green" title="E42">A1</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c6">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k</font>
 ;<br/>
<a NAME="E4:71_1_1_1"/>
<font color="Maroon" title="c6">c</font> is   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <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/>
<span class="kw">hence </span><a NAME="E5:71_1_1_1"/>
<font color="Maroon" title="c5">k</font> <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="c3">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:71_1_1_1"><span class="lab"><font color="Green" title="E42">A1</font></span></a>, <a class="txt" href="#E2:71_1_1"><span class="lab"><font color="Green" title="E40">A</font></span></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>
<a NAME="E4:71_1_1"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <span class="p2">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
;<br/>
<a NAME="E5:71_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">P2</font></span>: 
 ex <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
;<br/>
<span class="kw">consider </span><font color="Maroon" title="c5">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E7:71_1_1"/><span class="lab"><font color="Green" title="E43">B</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font>]
 <span class="kw">and </span><br/><a NAME="E8:71_1_1"/><span class="lab"><font color="Green" title="E44">C</font></span>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">k</font>
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S6" title="NAT_1:sch.6">NAT_1:sch 6</a>(<a class="txt" href="#E3:71_1_1"><span class="lab"><font color="Green" title="E41">P1</font></span></a>, <a class="txt" href="#E5:71_1_1"><span class="lab"><font color="Green" title="E42">P2</font></span></a>);<br/></span>
<span class="kw">consider </span><font color="Maroon" title="c6">c</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <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="c2">S</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E10:71_1_1"/><span class="lab"><font color="Green" title="E45">D</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c6">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:71_1_1"><span class="lab"><font color="Green" title="E43">B</font></span></a></span>;<br/>
<a NAME="E11:71_1_1"/>
for <font color="Olive" title="b1">D</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <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="c2">S</font></span>)</span> holds   <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">D</font> <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="c6">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:71_1_1"><span class="lab"><font color="Green" title="E45">D</font></span></a>, <a class="txt" href="#E8:71_1_1"><span class="lab"><font color="Green" title="E44">C</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:71_1_1"/>
 <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">S</font> is  <a href="dilworth.html#V3" title="DILWORTH:attr.3">with_finite_clique#</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="dilworth.html#D3" target="_self" title="DILWORTH:def.3">Lfheight</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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