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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>, <font color="Olive" title="b2">B</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="b3">e</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">M</font> holds <br/> (  <a href="matroid0.html#K4" title="MATROID0:func.4">Rnk</a> <font color="Olive" title="b1">A</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="Olive" title="b1">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b2">B</font></span>)</span> &amp;  <a href="matroid0.html#K4" title="MATROID0:func.4">Rnk</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">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="Olive" title="b3">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> <span class="p1">(<span class="default"><a href="matroid0.html#K4" title="MATROID0:func.4">Rnk</a> <font color="Olive" title="b1">A</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c3">B</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> holds <br/> (  <a href="matroid0.html#K4" title="MATROID0:func.4">Rnk</a> <font color="Maroon" title="c2">A</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> <font color="Maroon" title="c3">B</font></span>)</span> &amp;  <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="Olive" title="b1">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> <span class="p1">(<span class="default"><a href="matroid0.html#K4" title="MATROID0:func.4">Rnk</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">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"> (  <a href="matroid0.html#K4" title="MATROID0:func.4">Rnk</a> <font color="Maroon" title="c2">A</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> <font color="Maroon" title="c3">B</font></span>)</span> &amp;  <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="c4">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> <span class="p1">(<span class="default"><a href="matroid0.html#K4" title="MATROID0:func.4">Rnk</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )</span><br/>







<a NAME="E1:61"/><span class="lab"><font color="Green" title="E30">A1</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">e</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <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#T50" title="CARD_1:th.50">CARD_1:50</a></span>;<br/>
<span class="kw">thus </span><a NAME="E2:61"/>
 <a href="matroid0.html#K4" title="MATROID0:func.4">Rnk</a> <font color="Maroon" title="c2">A</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> <font color="Maroon" title="c3">B</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matroid0.html#T24" target="_self" title="MATROID0:th.24">Th24</a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <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="c4">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> <span class="p1">(<span class="default"><a href="matroid0.html#K4" title="MATROID0:func.4">Rnk</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span><br/>

<a NAME="E3:61"/><span class="lab"><font color="Green" title="E31">A2</font></span>: 
<span class="p1">(<span class="default"><a href="matroid0.html#K4" title="MATROID0:func.4">Rnk</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">e</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="matroid0.html#K4" title="MATROID0:func.4">Rnk</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">e</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="matroid0.html#K4" title="MATROID0:func.4">Rnk</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="matroid0.html#K4" title="MATROID0:func.4">Rnk</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">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="ref" href="matroid0.html#T25" target="_self" title="MATROID0:th.25">Th25</a></span>;<br/>
<a NAME="E4:61"/>
 <a href="matroid0.html#K4" title="MATROID0:func.4">Rnk</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">e</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <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> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">e</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<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/>
<a NAME="E5:61"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="matroid0.html#K4" title="MATROID0:func.4">Rnk</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="matroid0.html#K4" title="MATROID0:func.4">Rnk</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">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> <span class="p1">(<span class="default"><a href="matroid0.html#K4" title="MATROID0:func.4">Rnk</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61"><span class="lab"><font color="Green" title="E30">A1</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/>
<a NAME="E6:61"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A3</font></span>: 
<span class="p1">(<span class="default"><a href="matroid0.html#K4" title="MATROID0:func.4">Rnk</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">e</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="matroid0.html#K4" title="MATROID0:func.4">Rnk</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">e</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="matroid0.html#K4" title="MATROID0:func.4">Rnk</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E7:61"/>
 <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="c4">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> <span class="p1">(<span class="default"><a href="matroid0.html#K4" title="MATROID0:func.4">Rnk</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">e</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="matroid0.html#K4" title="MATROID0:func.4">Rnk</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">e</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:61"/>
 <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="c4">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> <span class="p1">(<span class="default"><a href="matroid0.html#K4" title="MATROID0:func.4">Rnk</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:61"><span class="lab"><font color="Green" title="E32">A3</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</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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