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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> holds   <a href="matroid0.html#K6" title="MATROID0:func.6">Span</a> <span class="p1">(<span class="default"><a href="matroid0.html#K6" title="MATROID0:func.6">Span</a> <font color="Olive" title="b1">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matroid0.html#K6" title="MATROID0:func.6">Span</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">  <a href="matroid0.html#K6" title="MATROID0:func.6">Span</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matroid0.html#K6" title="MATROID0:func.6">Span</a> <font color="Maroon" title="c2">A</font></span><br/>



<span class="kw">thus </span><a NAME="E1:72"/>
 <a href="matroid0.html#K6" title="MATROID0:func.6">Span</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> <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="comment"><font color="Red">:: according to </font></span><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> not  <a href="matroid0.html#K6" title="MATROID0:func.6">Span</a> <font color="Maroon" title="c2">A</font> <a href="matroid0.html#NR1" title="MATROID0:NR.1">c/=</a>  <a href="matroid0.html#K6" title="MATROID0:func.6">Span</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><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matroid0.html#K6" title="MATROID0:func.6">Span</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> or <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matroid0.html#K6" title="MATROID0:func.6">Span</a> <font color="Maroon" title="c2">A</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:72_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matroid0.html#K6" title="MATROID0:func.6">Span</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>
 ; <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">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matroid0.html#K6" title="MATROID0:func.6">Span</a> <font color="Maroon" title="c2">A</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">e</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">M</font><span class="kw"> such that </span><br/><a NAME="E3:72_1"/><span class="lab"><font color="Green" title="E37">A1</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">e</font>
 <span class="kw">and </span><br/><a NAME="E4:72_1"/><span class="lab"><font color="Green" title="E38">A2</font></span>: 
<font color="Maroon" title="c4">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>
 ;<br/>
<a NAME="E5:72_1"/>
<font color="Maroon" title="c4">e</font> <a href="matroid0.html#R2" title="MATROID0:pred.2">is_dependent_on</a> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:72_1"><span class="lab"><font color="Green" title="E38">A2</font></span></a>, <a class="ref" href="matroid0.html#T34" target="_self" title="MATROID0:th.34">Th34</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:72_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="txt" href="#E3:72_1"><span class="lab"><font color="Green" title="E37">A1</font></span></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>

<span class="kw">thus </span><a NAME="E2:72"/>
not  <a href="matroid0.html#K6" title="MATROID0:func.6">Span</a> <font color="Maroon" title="c2">A</font> <a href="matroid0.html#NR1" title="MATROID0:NR.1">c/=</a>  <a href="matroid0.html#K6" title="MATROID0:func.6">Span</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#T31" target="_self" title="MATROID0:th.31">Th31</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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