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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">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">M</font><br/> for <font color="Olive" title="b2">e</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">M</font>  st <font color="Olive" title="b2">e</font> <a href="matroid0.html#R2" title="MATROID0:pred.2">is_dependent_on</a>  <a href="matroid0.html#K6" title="MATROID0:func.6">Span</a> <font color="Olive" title="b1">A</font> holds <br/><font color="Olive" title="b2">e</font> <a href="matroid0.html#R2" title="MATROID0:pred.2">is_dependent_on</a> <font color="Olive" title="b1">A</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">A</font> be   <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"> for <font color="Olive" title="b1">e</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">M</font>  st <font color="Olive" title="b1">e</font> <a href="matroid0.html#R2" title="MATROID0:pred.2">is_dependent_on</a>  <a href="matroid0.html#K6" title="MATROID0:func.6">Span</a> <font color="Maroon" title="c2">A</font> holds <br/><font color="Olive" title="b1">e</font> <a href="matroid0.html#R2" title="MATROID0:pred.2">is_dependent_on</a> <font color="Maroon" title="c2">A</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">e</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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="c3">e</font> <a href="matroid0.html#R2" title="MATROID0:pred.2">is_dependent_on</a>  <a href="matroid0.html#K6" title="MATROID0:func.6">Span</a> <font color="Maroon" title="c2">A</font> implies <font color="Maroon" title="c3">e</font> <a href="matroid0.html#R2" title="MATROID0:pred.2">is_dependent_on</a> <font color="Maroon" title="c2">A</font> )</span><br/>





<span class="kw">assume </span><a NAME="E1:71"/><span class="lab"><font color="Green" title="E37">A1</font></span>: 
 <a href="matroid0.html#K4" title="MATROID0:func.4">Rnk</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matroid0.html#K6" title="MATROID0:func.6">Span</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">e</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matroid0.html#K4" title="MATROID0:func.4">Rnk</a> <span class="p1">(<span class="default"><a href="matroid0.html#K6" title="MATROID0:func.6">Span</a> <font color="Maroon" title="c2">A</font></span>)</span>
 ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="matroid0.html#D14" target="_self" title="MATROID0:def.14">MATROID0:def 14</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="c3">e</font> <a href="matroid0.html#R2" title="MATROID0:pred.2">is_dependent_on</a> <font color="Maroon" title="c2">A</font></span><br/>

<a NAME="E2:71"/><span class="lab"><font color="Green" title="E38">A2</font></span>: 
 <a href="matroid0.html#K4" title="MATROID0:func.4">Rnk</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matroid0.html#K4" title="MATROID0:func.4">Rnk</a> <span class="p1">(<span class="default"><a href="matroid0.html#K6" title="MATROID0:func.6">Span</a> <font color="Maroon" title="c2">A</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matroid0.html#T33" target="_self" title="MATROID0:th.33">Th33</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c4">Ca</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><span class="kw"> such that </span><br/><a NAME="E4:71"/><span class="lab"><font color="Green" title="E39">A3</font></span>: 
<font color="Maroon" title="c4">Ca</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">and </span><br/><a NAME="E5:71"/><span class="lab"><font color="Green" title="E40">A4</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c4">Ca</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">A</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/>
<a NAME="E6:71"/><span class="lab"><font color="Green" title="E41">A5</font></span>: 
 <a href="matroid0.html#K4" title="MATROID0:func.4">Rnk</a> <font color="Maroon" title="c2">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="c4">Ca</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:71"><span class="lab"><font color="Green" title="E40">A4</font></span></a>, <a class="ref" href="matroid0.html#T21" target="_self" title="MATROID0:th.21">Th21</a></span>;<br/>
<a NAME="E7:71"/><span class="lab"><font color="Green" title="E42">A6</font></span>: 
 <a href="matroid0.html#K4" title="MATROID0:func.4">Rnk</a> <font color="Maroon" title="c4">Ca</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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">e</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:71"><span class="lab"><font color="Green" title="E39">A3</font></span></a>, <a class="ref" href="matroid0.html#T24" target="_self" title="MATROID0:th.24">Th24</a>, <a class="ref" href="xboole_1.html#T10" title="XBOOLE_1:th.10">XBOOLE_1:10</a></span>;<br/>
<a NAME="E8:71"/>
<font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="matroid0.html#K6" title="MATROID0:func.6">Span</a> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matroid0.html#T31" target="_self" title="MATROID0:th.31">Th31</a></span>;<br/>
<a NAME="E9:71"/><span class="kw">then </span>
 <a href="matroid0.html#K4" title="MATROID0:func.4">Rnk</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">e</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <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">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:71"><span class="lab"><font color="Green" title="E37">A1</font></span></a>, <a class="txt" href="#E2:71"><span class="lab"><font color="Green" title="E38">A2</font></span></a>, <a class="ref" href="matroid0.html#T24" target="_self" title="MATROID0:th.24">Th24</a>, <a class="ref" href="xboole_1.html#T9" title="XBOOLE_1:th.9">XBOOLE_1:9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:71"/>
 <a href="matroid0.html#K4" title="MATROID0:func.4">Rnk</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">e</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <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">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:71"><span class="lab"><font color="Green" title="E41">A5</font></span></a>, <a class="txt" href="#E7:71"><span class="lab"><font color="Green" title="E42">A6</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="matroid0.html#D14" target="_self" title="MATROID0:def.14">MATROID0:def 14</a> <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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