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<span class="kw">let </span><font color="Maroon" title="c2">it1</font>, <font color="Maroon" title="c3">it2</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="firebrick">::  thesis: </font></span></a><span class="hide"> (  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#NM3" title="DILWORTH:NM.3">Clique-partition</a> of <font color="Maroon" title="c1">R</font> 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="c2">it1</font> &amp; (  for <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#NM3" title="DILWORTH:NM.3">Clique-partition</a> of <font color="Maroon" title="c1">R</font> holds  <font color="Maroon" title="c2">it1</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="Olive" title="b1">C</font> ) &amp;  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#NM3" title="DILWORTH:NM.3">Clique-partition</a> of <font color="Maroon" title="c1">R</font> 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="c3">it2</font> &amp; (  for <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#NM3" title="DILWORTH:NM.3">Clique-partition</a> of <font color="Maroon" title="c1">R</font> holds  <font color="Maroon" title="c3">it2</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="Olive" title="b1">C</font> ) implies <font color="Maroon" title="c2">it1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">it2</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:27_1_2"/><span class="lab"><font color="Green" title="E13">A5</font></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#NM3" title="DILWORTH:NM.3">Clique-partition</a> of <font color="Maroon" title="c1">R</font> 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="c2">it1</font>
 <span class="kw">and </span><br/><a NAME="E2:27_1_2"/><span class="lab"><font color="Green" title="E14">A6</font></span>: 
 for <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#NM3" title="DILWORTH:NM.3">Clique-partition</a> of <font color="Maroon" title="c1">R</font> holds  <font color="Maroon" title="c2">it1</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="Olive" title="b1">C</font>
 <span class="kw">and </span><br/><a NAME="E3:27_1_2"/><span class="lab"><font color="Green" title="E15">A7</font></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#NM3" title="DILWORTH:NM.3">Clique-partition</a> of <font color="Maroon" title="c1">R</font> 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="c3">it2</font>
 <span class="kw">and </span><br/><a NAME="E4:27_1_2"/><span class="lab"><font color="Green" title="E16">A8</font></span>: 
 for <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#NM3" title="DILWORTH:NM.3">Clique-partition</a> of <font color="Maroon" title="c1">R</font> holds  <font color="Maroon" title="c3">it2</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="Olive" title="b1">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="c2">it1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">it2</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c4">C1</font> being   <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 <font color="Maroon" title="c1">R</font><span class="kw"> such that </span><br/><a NAME="E6:27_1_2"/><span class="lab"><font color="Green" title="E17">A9</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c4">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">it1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27_1_2"><span class="lab"><font color="Green" title="E13">A5</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c5">C2</font> being   <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 <font color="Maroon" title="c1">R</font><span class="kw"> such that </span><br/><a NAME="E8:27_1_2"/><span class="lab"><font color="Green" title="E18">A10</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c5">C2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">it2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:27_1_2"><span class="lab"><font color="Green" title="E15">A7</font></span></a></span>;<br/>
<a NAME="E9:27_1_2"/>
( <font color="Maroon" title="c2">it1</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="c5">C2</font> &amp; <font color="Maroon" title="c3">it2</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="c4">C1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:27_1_2"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="txt" href="#E4:27_1_2"><span class="lab"><font color="Green" title="E16">A8</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:27_1_2"/>
<font color="Maroon" title="c2">it1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">it2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:27_1_2"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="txt" href="#E8:27_1_2"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</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/>


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