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<b>let </b><font color="Maroon" title="c1">F</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</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">V</font> being  <a href="vectsp_1.html#NM7" title="VECTSP_1:NM.7">VectSp</a> of <font color="Maroon" title="c1">F</font><br/> for <font color="Olive" title="b2">A</font> being  <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="Olive" title="b1">V</font></span>)</span> holds <br/> ( <font color="Olive" title="b2">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">independent</a> iff <font color="Olive" title="b2">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="Olive" title="b1">V</font> )</span><br/><b>let </b><font color="Maroon" title="c2">V</font> be   <a href="vectsp_1.html#NM7" title="VECTSP_1:NM.7">VectSp</a> of <font color="Maroon" title="c1">F</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">A</font> being  <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> holds <br/> ( <font color="Olive" title="b1">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">independent</a> iff <font color="Olive" title="b1">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><br/>



<b>set </b><font color="Maroon" title="c3">M</font> =  <a href="matroid0.html#K3" title="MATROID0:func.3">LinearlyIndependentSubsets</a> <font color="Maroon" title="c2">V</font>;<br/>
<b>let </b><font color="Maroon" title="c4">B</font> be   <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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">B</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">independent</a> iff <font color="Maroon" title="c4">B</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><br/>

<a NAME="E1:40"/>
 <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 href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">V</font> : <font color="Olive" title="b1">A</font> is <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a> </span>}</span> 
 
<b>by </b><i><a class="ref" href="matroid0.html#D9" target="_self" title="MATROID0:def.9">Def5</a></i>;<br/>
<a NAME="E2:40"/><b>then </b>
( <font color="Maroon" title="c4">B</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> iff  ex <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="c2">V</font> st <br/>( <font color="Maroon" title="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">A</font> &amp; <font color="Olive" title="b1">A</font> is <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a> ) )
 
;<br/>
<b>hence </b><a NAME="E3:40"/>
( <font color="Maroon" title="c4">B</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">independent</a> iff <font color="Maroon" title="c4">B</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> )
 <b>by </b><i><a class="ref" href="matroid0.html#D2" target="_self" title="MATROID0:def.2">Def8</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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