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<span class="kw">let </span><font color="Maroon" title="c3">A</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="matroid0.html#D5" target="_self" title="MATROID0:def.5">MATROID0:def 5</a>,<a class="ref" href="matroid0.html#D6" target="_self" title="MATROID0:def.6">MATROID0:def 6</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">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matroid0.html#K1" title="MATROID0:func.1">the_family_of</a> <span class="p1">(<span class="default"><a href="matroid0.html#K3" title="MATROID0:func.3">LinearlyIndependentSubsets</a> <font color="Maroon" title="c2">V</font></span>)</span> implies <font color="Maroon" title="c3">A</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c4">M</font> =  <a href="matroid0.html#K3" title="MATROID0:func.3">LinearlyIndependentSubsets</a> <font color="Maroon" title="c2">V</font>;<br/>
<span class="kw">assume </span><a NAME="E1:44_1_1"/>
<font color="Maroon" title="c3">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matroid0.html#K1" title="MATROID0:func.1">the_family_of</a> <span class="p1">(<span class="default"><a href="matroid0.html#K3" title="MATROID0:func.3">LinearlyIndependentSubsets</a> <font color="Maroon" title="c2">V</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">A</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> </span><br/>

<a NAME="E2:44_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">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 <span class="p1">(<span class="default"><a href="matroid0.html#K3" title="MATROID0:func.3">LinearlyIndependentSubsets</a> <font color="Maroon" title="c2">V</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matroid0.html#D2" target="_self" title="MATROID0:def.2">Def2</a></span>;<br/>
<a NAME="E3:44_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">A</font> is   <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">V</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matroid0.html#T11" target="_self" title="MATROID0:th.11">Th11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:44_1_1"/>
<font color="Maroon" title="c3">A</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_9.html#T25" title="VECTSP_9:th.25">VECTSP_9:25</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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