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<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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>  st <font color="Olive" title="b1">A</font> is <a href="matroid0.html#V5" title="MATROID0:attr.5">cycle</a> &amp; <font color="Olive" title="b2">B</font> is <a href="matroid0.html#V5" title="MATROID0:attr.5">cycle</a> &amp; <font color="Olive" title="b1">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">B</font> &amp; <font color="Olive" title="b3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b2">B</font> holds <br/> ex <font color="Olive" title="b4">C</font> being  <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="b4">C</font> is <a href="matroid0.html#V5" title="MATROID0:attr.5">cycle</a> &amp; <font color="Olive" title="b4">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><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><br/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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="Maroon" title="c2">A</font> is <a href="matroid0.html#V5" title="MATROID0:attr.5">cycle</a> &amp; <font color="Maroon" title="c3">B</font> is <a href="matroid0.html#V5" title="MATROID0:attr.5">cycle</a> &amp; <font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">B</font> holds <br/> ex <font color="Olive" title="b2">C</font> being  <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="b2">C</font> is <a href="matroid0.html#V5" title="MATROID0:attr.5">cycle</a> &amp; <font color="Olive" title="b2">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><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><br/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">A</font> is <a href="matroid0.html#V5" title="MATROID0:attr.5">cycle</a> &amp; <font color="Maroon" title="c3">B</font> is <a href="matroid0.html#V5" title="MATROID0:attr.5">cycle</a> &amp; <font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c4">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">B</font> implies  ex <font color="Olive" title="b1">C</font> being  <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">C</font> is <a href="matroid0.html#V5" title="MATROID0:attr.5">cycle</a> &amp; <font color="Olive" title="b1">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</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><br/>







<b>assume </b><b>that </b><br/><a NAME="E1:82"/><i><font color="Green" title="E44">Z0</font></i>: 
( <font color="Maroon" title="c2">A</font> is <a href="matroid0.html#V5" title="MATROID0:attr.5">cycle</a> &amp; <font color="Maroon" title="c3">B</font> is <a href="matroid0.html#V5" title="MATROID0:attr.5">cycle</a> &amp; <font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c4">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">B</font> )
 <b>and </b><br/><a NAME="E2:82"/><i><font color="Green" title="E45">Z1</font></i>: 
for <font color="Olive" title="b1">C</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">M</font>  st <font color="Olive" title="b1">C</font> is <a href="matroid0.html#V5" title="MATROID0:attr.5">cycle</a> holds <br/><font color="Olive" title="b1">C</font> <a href="matroid0.html#NR1" title="MATROID0:NR.1">c/=</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> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</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 class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E3:82"/><i><font color="Green" title="E46">A0</font></i>: 
( <font color="Maroon" title="c4">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c4">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> )
 
<b>by </b><i><a class="txt" href="#E1:82"><i><font color="Green" title="E44">Z0</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">Ae</font> = <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</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>, <font color="Maroon" title="c6">Be</font> = <font color="Maroon" title="c3">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</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> as  <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> <b>by </b><i><a class="txt" href="#E1:82"><i><font color="Green" title="E44">Z0</font></i></a>, <a class="ref" href="matroid0.html#D16" target="_self" title="MATROID0:def.16">CYCLE</a></i>;<br/>
<a NAME="E5:82"/><i><font color="Green" title="E47">A1</font></i>: 
( <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">Ae</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</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> &amp; <font color="Maroon" title="c3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">Be</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</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> )
 
<b>by </b><i><a class="txt" href="#E3:82"><i><font color="Green" title="E46">A0</font></i></a>, <a class="ref" href="zfmisc_1.html#T140" title="ZFMISC_1:th.140">ZFMISC_1:140</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">A'</font> = <font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c8">B'</font> = <font color="Maroon" title="c3">B</font> as  <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> <b>by </b><i><a class="txt" href="#E1:82"><i><font color="Green" title="E44">Z0</font></i></a></i>;<br/>
<a NAME="E7:82"/><i><font color="Green" title="E48">A3</font></i>: 
<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> <font color="Maroon" title="c3">B</font></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> <font color="Maroon" title="c3">B</font></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> <font color="Maroon" title="c3">B</font></span>)</span>
 
<b>by </b><i><a class="ref" href="matroid0.html#T25" target="_self" title="MATROID0:th.25">R3</a></i>;<br/>
<a NAME="E8:82"/><i><font color="Green" title="E49">B2</font></i>: 
<font color="Maroon" title="c4">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E9:82"/><b>then </b>
( <font color="Maroon" title="c4">e</font> <a href="aofa_000.html#NR1" title="AOFA_000:NR.1">nin</a> <font color="Maroon" title="c5">Ae</font> &amp; <font color="Maroon" title="c4">e</font> <a href="aofa_000.html#NR1" title="AOFA_000:NR.1">nin</a> <font color="Maroon" title="c6">Be</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>;<br/>
<a NAME="E10:82"/><b>then </b><i><font color="Green" title="E50">A4</font></i>: 
(  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c7">A'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c5">Ae</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c8">B'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c6">Be</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 
<b>by </b><i><a class="txt" href="#E5:82"><i><font color="Green" title="E47">A1</font></i></a>, <a class="ref" href="card_2.html#T54" title="CARD_2:th.54">CARD_2:54</a></i>;<br/>
<a NAME="E11:82"/><b>then </b>
( <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 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c5">Ae</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1">(<span class="default"><a href="matroid0.html#K4" title="MATROID0:func.4">Rnk</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c6">Be</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 
<b>by </b><i><a class="txt" href="#E1:82"><i><font color="Green" title="E44">Z0</font></i></a>, <a class="ref" href="matroid0.html#T39" target="_self" title="MATROID0:th.39">C4</a></i>;<br/>
<b>then </b><i><font color="Green" title="E51">A5</font></i>: <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">A'</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c8">B'</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">A'</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c8">B'</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<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>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matroid0.html#K4" title="MATROID0:func.4">Rnk</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="#E10:82"><i><font color="Green" title="E50">A4</font></i></a>, <a class="ref" href="hallmar1.html#T1" title="HALLMAR1:th.1">HALLMAR1:1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="p3">(<span class="default"><a href="matroid0.html#K4" title="MATROID0:func.4">Rnk</a> <font color="Maroon" title="c3">B</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1

;<br/>
<a NAME="E13:82"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matroid0.html#K4" title="MATROID0:func.4">Rnk</a> <span class="p3">(<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>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="matroid0.html#K4" title="MATROID0:func.4">Rnk</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">B</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<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="p2">(<span class="default"><a href="matroid0.html#K4" title="MATROID0:func.4">Rnk</a> <font color="Maroon" title="c3">B</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="#E7:82"><i><font color="Green" title="E48">A3</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<a NAME="E14:82"/><b>then </b><i><font color="Green" title="E52">A7</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matroid0.html#K4" title="MATROID0:func.4">Rnk</a> <span class="p4">(<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>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><a href="matroid0.html#K4" title="MATROID0:func.4">Rnk</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">B</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">A'</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c8">B'</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">A'</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c8">B'</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E12:82"><i><font color="Green" title="E51">A5</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<a NAME="E15:82"/><i><font color="Green" title="E53">A9</font></i>: 
( <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">B</font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E16:82"/>
<font color="Maroon" title="c2">A</font> <a href="matroid0.html#NR1" title="MATROID0:NR.1">c/=</a> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">B</font>
 
<b>by </b><i><a class="txt" href="#E1:82"><i><font color="Green" title="E44">Z0</font></i></a>, <a class="ref" href="matroid0.html#T41" target="_self" title="MATROID0:th.41">C1</a>, <a class="txt" href="#E15:82"><i><font color="Green" title="E53">A9</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c9">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E18:82"/><i><font color="Green" title="E54">B0</font></i>: 
( <font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c9">a</font> <a href="aofa_000.html#NR1" title="AOFA_000:NR.1">nin</a> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">B</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">a</font> = <font color="Maroon" title="c9">a</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">M</font> <b>by </b><i><a class="txt" href="#E18:82"><i><font color="Green" title="E54">B0</font></i></a></i>;<br/>
<a NAME="E20:82"/>
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c10">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">B</font>
 
<b>by </b><i><a class="txt" href="#E18:82"><i><font color="Green" title="E54">B0</font></i></a>, <a class="ref" href="zfmisc_1.html#T56" title="ZFMISC_1:th.56">ZFMISC_1:56</a></i>;<br/>
<a NAME="E21:82"/><b>then </b>
( <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c10">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c10">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">independent</a> )
 
<b>by </b><i><a class="txt" href="#E1:82"><i><font color="Green" title="E44">Z0</font></i></a>, <a class="txt" href="#E15:82"><i><font color="Green" title="E53">A9</font></i></a>, <a class="txt" href="#E18:82"><i><font color="Green" title="E54">B0</font></i></a>, <a class="ref" href="matroid0.html#D16" target="_self" title="MATROID0:def.16">CYCLE</a>, <a class="ref" href="xboole_1.html#T86" title="XBOOLE_1:th.86">XBOOLE_1:86</a></i>;<br/>
<a NAME="E22:82"/><b>then </b>
<font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">B</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">independent</a>
 
<b>by </b><i><a class="ref" href="matroid0.html#T3" target="_self" title="MATROID0:th.3">M1</a></i>;<br/>
<a NAME="E23:82"/><b>then </b>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">A'</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c8">B'</font></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"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">B</font></span>)</span>
 
<b>by </b><i><a class="ref" href="matroid0.html#T21" target="_self" title="MATROID0:th.21">ThR4</a></i>;<br/>
<a NAME="E24:82"/><b>then </b><i><font color="Green" title="E55">Z2</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matroid0.html#K4" title="MATROID0:func.4">Rnk</a> <span class="p4">(<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>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><a href="matroid0.html#K4" title="MATROID0:func.4">Rnk</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">B</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matroid0.html#K4" title="MATROID0:func.4">Rnk</a> <span class="p4">(<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>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">A'</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c8">B'</font></span>)</span></span>)</span>
 
;<br/>
<a NAME="E25:82"/>
for <font color="Olive" title="b1">C</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">M</font>  st <font color="Olive" title="b1">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</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> holds <br/> not <font color="Olive" title="b1">C</font> is <a href="matroid0.html#V5" title="MATROID0:attr.5">cycle</a>
 
<b>by </b><i><a class="txt" href="#E2:82"><i><font color="Green" title="E45">Z1</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c11">C</font> = <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> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</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> as  <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> <b>by </b><i><a class="ref" href="matroid0.html#T42" target="_self" title="MATROID0:th.42">C3</a></i>;<br/>
<a NAME="E27:82"/><i><font color="Green" title="E56">B1</font></i>: 
( <font color="Maroon" title="c4">e</font> <a href="aofa_000.html#NR1" title="AOFA_000:NR.1">nin</a> <font color="Maroon" title="c11">C</font> &amp; <font color="Maroon" title="c4">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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> )
 
<b>by </b><i><a class="txt" href="#E8:82"><i><font color="Green" title="E49">B2</font></i></a>, <a class="txt" href="#E3:82"><i><font color="Green" title="E46">A0</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>;<br/>
<a NAME="E28:82"/><b>then </b><i><font color="Green" title="E57">B3</font></i>: 
<font color="Maroon" title="c11">C</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</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> <font color="Maroon" title="c7">A'</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c8">B'</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T140" title="ZFMISC_1:th.140">ZFMISC_1:140</a></i>;<br/>
<a NAME="E29:82"/>
<font color="Maroon" title="c11">C</font> <a href="matroid0.html#R1" title="MATROID0:pred.1">is_maximal_independent_in</a> <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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:82_2"/>
( <font color="Maroon" title="c11">C</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">independent</a> &amp; <font color="Maroon" title="c11">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <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> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="matroid0.html#D10" target="_self" title="MATROID0:def.10">MATROID0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">B</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">M</font>  st <font color="Olive" title="b1">B</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">independent</a> &amp; <font color="Olive" title="b1">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <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> &amp; <font color="Maroon" title="c11">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">B</font> holds <br/><font color="Maroon" title="c11">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">B</font></span><br/>

<b>let </b><font color="Maroon" title="c12">D</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c12">D</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">independent</a> &amp; <font color="Maroon" title="c12">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <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> &amp; <font color="Maroon" title="c11">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">D</font> implies <font color="Maroon" title="c11">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">D</font> )</span><br/>

<a NAME="E2:82_2"/>
( not <font color="Maroon" title="c2">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">independent</a> &amp; <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <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> )
 
<b>by </b><i><a class="txt" href="#E1:82"><i><font color="Green" title="E44">Z0</font></i></a>, <a class="ref" href="matroid0.html#D16" target="_self" title="MATROID0:def.16">CYCLE</a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E3:82_2"/><b>then </b>
not <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> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">independent</a>
 
<b>by </b><i><a class="ref" href="matroid0.html#T3" target="_self" title="MATROID0:th.3">M1</a></i>;<br/>
<b>hence </b><a NAME="E4:82_2"/>
( <font color="Maroon" title="c12">D</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">independent</a> &amp; <font color="Maroon" title="c12">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <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> &amp; <font color="Maroon" title="c11">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">D</font> implies <font color="Maroon" title="c11">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">D</font> )
 <b>by </b><i><a class="txt" href="#E28:82"><i><font color="Green" title="E57">B3</font></i></a>, <a class="ref" href="borsuk_5.html#T2" title="BORSUK_5:th.2">BORSUK_5:2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>then </b><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> <font color="Maroon" title="c3">B</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 = 
<span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c11">C</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
<b>by </b><i><a class="ref" href="matroid0.html#T19" target="_self" title="MATROID0:th.19">ThR2</a></i>
<br/>.= 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">A'</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c8">B'</font></span>)</span>
<b>by </b><i><a class="txt" href="#E27:82"><i><font color="Green" title="E56">B1</font></i></a>, <a class="txt" href="#E28:82"><i><font color="Green" title="E57">B3</font></i></a>, <a class="ref" href="card_2.html#T54" title="CARD_2:th.54">CARD_2:54</a></i>
;<br/>
<b>hence </b><a NAME="E31:82"/>
contradiction
 <b>by </b><i><a class="txt" href="#E24:82"><i><font color="Green" title="E55">Z2</font></i></a>, <a class="txt" href="#E14:82"><i><font color="Green" title="E52">A7</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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