<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">M</font> be  non <a href="pencil_1.html#V3" title="PENCIL_1:attr.3">void</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"> ( <font color="Maroon" title="c1">M</font> is <a href="matroid0.html#V1" title="MATROID0:attr.1">subset-closed</a> iff 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>  st <font color="Olive" title="b1">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">independent</a> &amp; <font color="Olive" title="b2">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">A</font> holds <br/><font color="Olive" title="b2">B</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">independent</a> )</span><br/>

<b>thus </b><a NAME="E1:15"/>
( <font color="Maroon" title="c1">M</font> is <a href="matroid0.html#V1" title="MATROID0:attr.1">subset-closed</a> implies 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>  st <font color="Olive" title="b1">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">independent</a> &amp; <font color="Olive" title="b2">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">A</font> holds <br/><font color="Olive" title="b2">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#T1" target="_self" title="MATROID0:th.1">Th2</a></i>; <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>  st <font color="Olive" title="b1">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">independent</a> &amp; <font color="Olive" title="b2">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">A</font> holds <br/><font color="Olive" title="b2">B</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">independent</a> ) implies <font color="Maroon" title="c1">M</font> is <a href="matroid0.html#V1" title="MATROID0:attr.1">subset-closed</a> )</span><br/>

<b>assume </b><a NAME="E2:15"/><i><font color="Green" title="E6">03</font></i>: 
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>  st <font color="Olive" title="b1">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">independent</a> &amp; <font color="Olive" title="b2">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">A</font> holds <br/><font color="Olive" title="b2">B</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">independent</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">M</font> is <a href="matroid0.html#V1" title="MATROID0:attr.1">subset-closed</a></span><br/>

<b>let </b><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="classes1.html#D1" title="CLASSES1:def.1">CLASSES1:def 1</a>,<a class="ref" href="matroid0.html#D3" target="_self" title="MATROID0:def.3">MATROID0:def 3</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="c2">x</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> <font color="Maroon" title="c1">M</font> or <font color="Maroon" title="c3">y</font> <a href="matroid0.html#NR1" title="MATROID0:NR.1">c/=</a> <font color="Maroon" title="c2">x</font> or <font color="Maroon" title="c3">y</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> <font color="Maroon" title="c1">M</font> )</span><br/>

<b>assume </b><a NAME="E3:15"/>
<font color="Maroon" title="c2">x</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> <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="c3">y</font> <a href="matroid0.html#NR1" title="MATROID0:NR.1">c/=</a> <font color="Maroon" title="c2">x</font> or <font color="Maroon" title="c3">y</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> <font color="Maroon" title="c1">M</font> )</span><br/>

<a NAME="E4:15"/><b>then </b><i><font color="Green" title="E7">04</font></i>: 
<font color="Maroon" title="c2">x</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 <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>;<br/>
<b>assume </b><a NAME="E5:15"/>
<font color="Maroon" title="c3">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">y</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> <font color="Maroon" title="c1">M</font></span><br/>

<a NAME="E6:15"/><b>then </b>
<font color="Maroon" title="c3">y</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 <font color="Maroon" title="c1">M</font>
 
<b>by </b><i><a class="txt" href="#E2:15"><i><font color="Green" title="E6">03</font></i></a>, <a class="txt" href="#E4:15"><i><font color="Green" title="E7">04</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E7:15"/>
<font color="Maroon" title="c3">y</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> <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>
