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<span class="kw">let </span><font color="Maroon" title="c1">M</font> be   <a href="matroid0.html#V4" title="MATROID0:attr.4">finite-degree</a>  <a href="matroid0.html#NM2" title="MATROID0:NM.2">Matroid</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">C</font> being  <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">M</font> holds <br/> ( <font color="Olive" title="b1">C</font> is  <a href="matroid0.html#NV1" title="MATROID0:NV.1">independent</a>  iff  <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>  <a href="matroid0.html#K4" title="MATROID0:func.4">Rnk</a> <font color="Olive" title="b1">C</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">C</font> be   <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">M</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="matroid0.html#NV1" title="MATROID0:NV.1">independent</a>  iff  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matroid0.html#K4" title="MATROID0:func.4">Rnk</a> <font color="Maroon" title="c2">C</font> )</span><br/>



<span class="kw">set </span><font color="Maroon" title="c3">X</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">A</font></span>)</span> where <font color="Olive" title="b1">A</font> is   <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">independent</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">M</font> : <font color="Olive" title="b1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">C</font> </span>}</span> ;<br/>
<div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matroid0.html#K4" title="MATROID0:func.4">Rnk</a> <font color="Maroon" title="c2">C</font> implies <font color="Maroon" title="c2">C</font> is  <a href="matroid0.html#NV1" title="MATROID0:NV.1">independent</a>  )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:54_1"/>
<font color="Maroon" title="c2">C</font> is  <a href="matroid0.html#NV1" title="MATROID0:NV.1">independent</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matroid0.html#K4" title="MATROID0:func.4">Rnk</a> <font color="Maroon" title="c2">C</font></span><br/><a NAME="E2:54_1"/><span class="kw">then </span>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">A</font></span>)</span> where <font color="Olive" title="b1">A</font> is   <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">independent</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">M</font> : <font color="Olive" title="b1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">C</font> </span>}</span> 
 ;<br/><a NAME="E3:54_1"/><span class="kw">then </span>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="matroid0.html#K4" title="MATROID0:func.4">Rnk</a> <font color="Maroon" title="c2">C</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></span>;<br/><a NAME="E4:54_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A1</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">C</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matroid0.html#K4" title="MATROID0:func.4">Rnk</a> <font color="Maroon" title="c2">C</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T40" title="NAT_1:th.40">NAT_1:40</a></span>;<br/><a NAME="E5:54_1"/>
 <a href="matroid0.html#K4" title="MATROID0:func.4">Rnk</a> <font color="Maroon" title="c2">C</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="c2">C</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matroid0.html#T20" target="_self" title="MATROID0:th.20">Th20</a></span>;<br/><span class="kw">hence </span><a NAME="E6:54_1"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matroid0.html#K4" title="MATROID0:func.4">Rnk</a> <font color="Maroon" title="c2">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:54_1"><span class="lab"><font color="Green" title="E26">A1</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="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<a NAME="E2:54"/>
 ex <font color="Olive" title="b1">A</font> being  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">independent</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">M</font> st <br/>( <font color="Olive" title="b1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">C</font> &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matroid0.html#K4" title="MATROID0:func.4">Rnk</a> <font color="Maroon" title="c2">C</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matroid0.html#T18" target="_self" title="MATROID0:th.18">Th18</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:54"/>
(  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matroid0.html#K4" title="MATROID0:func.4">Rnk</a> <font color="Maroon" title="c2">C</font> implies <font color="Maroon" title="c2">C</font> is  <a href="matroid0.html#NV1" title="MATROID0:NV.1">independent</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_fin.html#T1" title="CARD_FIN:th.1">CARD_FIN:1</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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