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<b>let </b><font color="Maroon" title="c1">M</font> be  <a href="matroid0.html#V1" title="MATROID0:attr.1">subset-closed</a> <a href="matroid0.html#NM1" title="MATROID0:NM.1">SubsetFamilyStr</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( not <font color="Maroon" title="c1">M</font> is <a href="pencil_1.html#V3" title="PENCIL_1:attr.3">void</a> iff  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matroid0.html#K1" title="MATROID0:func.1">the_family_of</a> <font color="Maroon" title="c1">M</font> )</span><br/>

<div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matroid0.html#K1" title="MATROID0:func.1">the_family_of</a> <font color="Maroon" title="c1">M</font> implies  not <font color="Maroon" title="c1">M</font> is <a href="pencil_1.html#V3" title="PENCIL_1:attr.3">void</a> )</span>
<div class="add"><b>assume </b><a NAME="E1:13_1"/>
not <font color="Maroon" title="c1">M</font> is <a href="pencil_1.html#V3" title="PENCIL_1:attr.3">void</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matroid0.html#K1" title="MATROID0:func.1">the_family_of</a> <font color="Maroon" title="c1">M</font></span><br/><b>then reconsider </b><font color="Maroon" title="c2">M'</font> = <font color="Maroon" title="c1">M</font> as  non <a href="pencil_1.html#V3" title="PENCIL_1:attr.3">void</a> <a href="matroid0.html#V1" title="MATROID0:attr.1">subset-closed</a> <a href="matroid0.html#NM1" title="MATROID0:NM.1">SubsetFamilyStr</a> ;<br/><b>consider </b><font color="Maroon" title="c3">a</font> being  <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="c2">M'</font>;<br/><a NAME="E4:13_1"/>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">a</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</a></i>;<br/><a NAME="E5:13_1"/><b>then </b>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  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 <font color="Maroon" title="c2">M'</font>
 <b>by </b><i><a class="ref" href="matroid0.html#T1" target="_self" title="MATROID0:th.1">Th2</a></i>;<br/><b>hence </b><a NAME="E6:13_1"/>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matroid0.html#K1" title="MATROID0:func.1">the_family_of</a> <font color="Maroon" title="c1">M</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/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:13"/>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matroid0.html#K1" title="MATROID0:func.1">the_family_of</a> <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"> not <font color="Maroon" title="c1">M</font> is <a href="pencil_1.html#V3" title="PENCIL_1:attr.3">void</a></span><br/>

<b>hence </b><a NAME="E3:13"/>
not the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">M</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="pencil_1.html#D4" title="PENCIL_1:def.4">PENCIL_1:def 4</a> <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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