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

<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/>
<a NAME="E1:43_1_1"/><i><font color="Green" title="E18">A2</font></i>: 
 <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/>
<b>let </b><font color="Maroon" title="c4">A</font>, <font color="Maroon" title="c5">B</font> be  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</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>; <i><font color="Red">:: according to </font></i><a class="ref" href="matroid0.html#D4" target="_self" title="MATROID0:def.4">MATROID0:def 4</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">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> &amp; <font color="Maroon" title="c5">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> &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c5">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c4">A</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies  ex <font color="Olive" title="b1">e</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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> st <br/>( <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c4">A</font> &amp; <font color="Maroon" title="c4">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">e</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <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> ) )</span><br/>

<b>assume </b><a NAME="E2:43_1_1"/><i><font color="Green" title="E19">A5</font></i>: 
( <font color="Maroon" title="c4">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> &amp; <font color="Maroon" title="c5">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> &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c5">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c4">A</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">e</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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> st <br/>( <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c4">A</font> &amp; <font color="Maroon" title="c4">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">e</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <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> )</span><br/>

<a NAME="E3:43_1_1"/>
( <font color="Maroon" title="c4">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">independent</a> &amp; <font color="Maroon" title="c5">B</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">independent</a> )
 
<b>by </b><i><a class="txt" href="#E2:43_1_1"><i><font color="Green" title="E19">A5</font></i></a>, <a class="ref" href="matroid0.html#D2" target="_self" title="MATROID0:def.2">Def8</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c6">A'</font> = <font color="Maroon" title="c4">A</font>, <font color="Maroon" title="c7">B'</font> = <font color="Maroon" title="c5">B</font> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <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#T11" target="_self" title="MATROID0:th.11">Th21</a></i>;<br/>
<b>set </b><font color="Maroon" title="c8">V'</font> =  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">A'</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">B'</font></span>)</span>;<br/>
<a NAME="E5:43_1_1"/>
<font color="Maroon" title="c6">A'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">A'</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">B'</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">a</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="tarski.html#D3" title="TARSKI:def.3">TARSKI: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="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">A'</font> or <font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">A'</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">B'</font></span>)</span></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:43_1_1_1"/>
<font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">A'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">A'</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">B'</font></span>)</span></span>)</span></span><br/>

<a NAME="E2:43_1_1_1"/><b>then </b>
<font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">A'</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">B'</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E3:43_1_1_1"/><b>then </b>
<font color="Maroon" title="c9">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">A'</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">B'</font></span>)</span>
 
<b>by </b><i><a class="ref" href="vectsp_7.html#T13" title="VECTSP_7:th.13">VECTSP_7:13</a></i>;<br/>
<b>hence </b><a NAME="E4:43_1_1_1"/>
<font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">A'</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">B'</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</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>then reconsider </b><font color="Maroon" title="c9">A''</font> = <font color="Maroon" title="c6">A'</font> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <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 <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">A'</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">B'</font></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="vectsp_9.html#T16" title="VECTSP_9:th.16">VECTSP_9:16</a></i>;<br/>
<a NAME="E7:43_1_1"/>
<font color="Maroon" title="c7">B'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">A'</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">B'</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">a</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="tarski.html#D3" title="TARSKI:def.3">TARSKI: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="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B'</font> or <font color="Maroon" title="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">A'</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">B'</font></span>)</span></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:43_1_1_2"/>
<font color="Maroon" title="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">A'</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">B'</font></span>)</span></span>)</span></span><br/>

<a NAME="E2:43_1_1_2"/><b>then </b>
<font color="Maroon" title="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">A'</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">B'</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E3:43_1_1_2"/><b>then </b>
<font color="Maroon" title="c10">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">A'</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">B'</font></span>)</span>
 
<b>by </b><i><a class="ref" href="vectsp_7.html#T13" title="VECTSP_7:th.13">VECTSP_7:13</a></i>;<br/>
<b>hence </b><a NAME="E4:43_1_1_2"/>
<font color="Maroon" title="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">A'</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">B'</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</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>then reconsider </b><font color="Maroon" title="c10">B''</font> = <font color="Maroon" title="c7">B'</font> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <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 <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">A'</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">B'</font></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="vectsp_9.html#T16" title="VECTSP_9:th.16">VECTSP_9:16</a></i>;<br/>
<a NAME="E9:43_1_1"/><i><font color="Green" title="E20">A0</font></i>: 
 <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">A'</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">B'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">A''</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c10">B''</font></span>)</span>
 
<b>by </b><i><a class="ref" href="vectsp_9.html#T21" title="VECTSP_9:th.21">VECTSP_9:21</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c11">D</font> being    <a href="vectsp_7.html#M1" title="VECTSP_7:mode.1">Basis</a> of  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">A'</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">B'</font></span>)</span><b> such that </b><br/><a NAME="E11:43_1_1"/><i><font color="Green" title="E21">A4</font></i>: 
<font color="Maroon" title="c7">B'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">D</font>
 <b>by </b><i><a class="ref" href="vectsp_7.html#T27" title="VECTSP_7:th.27">VECTSP_7:27</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">C</font> being    <a href="vectsp_7.html#M1" title="VECTSP_7:mode.1">Basis</a> of  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">A'</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">B'</font></span>)</span><b> such that </b><br/><a NAME="E13:43_1_1"/><i><font color="Green" title="E22">A3</font></i>: 
<font color="Maroon" title="c12">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">A''</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c10">B''</font>
 <b>by </b><i><a class="txt" href="#E9:43_1_1"><i><font color="Green" title="E20">A0</font></i></a>, <a class="ref" href="vectsp_7.html#T28" title="VECTSP_7:th.28">VECTSP_7:28</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c13">e</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c12">C</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c6">A'</font></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">c</font> = <font color="Maroon" title="c12">C</font> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="txt" href="#E13:43_1_1"><i><font color="Green" title="E22">A3</font></i></a></i>;<br/>
<a NAME="E16:43_1_1"/><i><font color="Green" title="E23">A8</font></i>: 
( <font color="Maroon" title="c14">c</font> is    <a href="vectsp_7.html#M1" title="VECTSP_7:mode.1">Basis</a> of  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">A'</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">B'</font></span>)</span> &amp; <font color="Maroon" title="c6">A'</font> is    <a href="vectsp_7.html#M1" title="VECTSP_7:mode.1">Basis</a> of  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c6">A'</font> )
 
<b>by </b><i><a class="ref" href="ranknull.html#T20" title="RANKNULL:th.20">RANKNULL:20</a></i>;<br/>
<a NAME="E17:43_1_1"/><b>then </b><i><font color="Green" title="E24">BB</font></i>: 
(  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">A'</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">B'</font></span>)</span> is <a href="matrlin.html#V2" title="MATRLIN:attr.2">finite-dimensional</a> &amp;  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c6">A'</font> is <a href="matrlin.html#V2" title="MATRLIN:attr.2">finite-dimensional</a> )
 
<b>by </b><i><a class="ref" href="matrlin.html#D3" title="MATRLIN:def.3">MATRLIN:def 3</a></i>;<br/>
<a NAME="E18:43_1_1"/><b>then </b>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c14">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c11">D</font>
 
<b>by </b><i><a class="ref" href="vectsp_9.html#T26" title="VECTSP_9:th.26">VECTSP_9:26</a></i>;<br/>
<a NAME="E19:43_1_1"/><b>then </b>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c5">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c14">c</font>
 
<b>by </b><i><a class="txt" href="#E11:43_1_1"><i><font color="Green" title="E21">A4</font></i></a>, <a class="ref" href="card_1.html#T27" title="CARD_1:th.27">CARD_1:27</a></i>;<br/>
<a NAME="E20:43_1_1"/><b>then </b>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c5">B</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c14">c</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T40" title="NAT_1:th.40">NAT_1:40</a></i>;<br/>
<a NAME="E21:43_1_1"/><b>then </b><i><font color="Green" title="E25">A9</font></i>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c4">A</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c14">c</font>
 
<b>by </b><i><a class="txt" href="#E2:43_1_1"><i><font color="Green" title="E19">A5</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:43_1_1_3"/>
<font color="Maroon" title="c12">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c6">A'</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><b>then reconsider </b><font color="Maroon" title="c15">C'</font> = <font color="Maroon" title="c12">C</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c6">A'</font></span>)</span> ;<br/><a NAME="E3:43_1_1_3"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c6">A'</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V</font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#D2" title="VECTSP_4:def.2">VECTSP_4:def 2</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c16">C''</font> = <font color="Maroon" title="c15">C'</font> as   <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="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E5:43_1_1_3"/>
<font color="Maroon" title="c12">C</font> is <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a>
 <b>by </b><i><a class="ref" href="vectsp_7.html#D3" title="VECTSP_7:def.3">VECTSP_7:def 3</a></i>;<br/><a NAME="E6:43_1_1_3"/><b>then </b>
<font color="Maroon" title="c16">C''</font> is <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a>
 <b>by </b><i><a class="ref" href="vectsp_9.html#T15" title="VECTSP_9:th.15">VECTSP_9:15</a></i>;<br/><b>then consider </b><font color="Maroon" title="c17">E</font> being    <a href="vectsp_7.html#M1" title="VECTSP_7:mode.1">Basis</a> of  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c6">A'</font><b> such that </b><br/><a NAME="E8:43_1_1_3"/><i><font color="Green" title="E26">B2</font></i>: 
<font color="Maroon" title="c15">C'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c17">E</font>
 <b>by </b><i><a class="ref" href="vectsp_7.html#T27" title="VECTSP_7:th.27">VECTSP_7:27</a>, <a class="ref" href="vectsp_9.html#T16" title="VECTSP_9:th.16">VECTSP_9:16</a></i>;<br/><a NAME="E9:43_1_1_3"/><i><font color="Green" title="E27">B3</font></i>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c17">E</font>
 <b>by </b><i><a class="txt" href="#E16:43_1_1"><i><font color="Green" title="E23">A8</font></i></a>, <a class="txt" href="#E17:43_1_1"><i><font color="Green" title="E24">BB</font></i></a>, <a class="ref" href="vectsp_9.html#T26" title="VECTSP_9:th.26">VECTSP_9:26</a></i>;<br/><a NAME="E10:43_1_1_3"/><b>then </b>
<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c17">E</font> <a href="tarski.html#R2" title="TARSKI:pred.2">are_equipotent</a> 
 <b>by </b><i><a class="ref" href="card_1.html#T21" title="CARD_1:th.21">CARD_1:21</a></i>;<br/><a NAME="E11:43_1_1_3"/><b>then </b>
<font color="Maroon" title="c17">E</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="ref" href="card_1.html#T68" title="CARD_1:th.68">CARD_1:68</a></i>;<br/><b>hence </b><a NAME="E12:43_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="#E21:43_1_1"><i><font color="Green" title="E25">A9</font></i></a>, <a class="txt" href="#E8:43_1_1_3"><i><font color="Green" title="E26">B2</font></i></a>, <a class="txt" href="#E9:43_1_1_3"><i><font color="Green" title="E27">B3</font></i></a>, <a class="ref" href="nat_1.html#T44" title="NAT_1:th.44">NAT_1:44</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>then consider </b><font color="Maroon" title="c15">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E24:43_1_1"/><i><font color="Green" title="E26">AA</font></i>: 
( <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">C</font> &amp; <font color="Maroon" title="c15">x</font> <a href="aofa_000.html#NR1" title="AOFA_000:NR.1">nin</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c6">A'</font></span>)</span> )
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E25:43_1_1"/>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">C</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c6">A'</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E24:43_1_1"><i><font color="Green" title="E26">AA</font></i></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>;<br/>
<a NAME="E26:43_1_1"/><b>then </b><i><font color="Green" title="E27">A6</font></i>: 
( <font color="Maroon" title="c13">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">C</font> &amp; <font color="Maroon" title="c13">e</font> <a href="aofa_000.html#NR1" title="AOFA_000:NR.1">nin</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c6">A'</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>;<br/>
<a NAME="E27:43_1_1"/><b>then </b><i><font color="Green" title="E28">A7</font></i>: 
<font color="Maroon" title="c13">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">B</font>
 
<b>by </b><i><a class="txt" href="#E13:43_1_1"><i><font color="Green" title="E22">A3</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c16">e</font> = <font color="Maroon" title="c13">e</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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> ;<br/>
<b>take </b>
<font color="Maroon" title="c16">e</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c16">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c4">A</font> &amp; <font color="Maroon" title="c4">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c16">e</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <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> )</span><br/>

<a NAME="E29:43_1_1"/>
<font color="Maroon" title="c4">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c6">A'</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c17">x</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="tarski.html#D3" title="TARSKI:def.3">TARSKI: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="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> or <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c6">A'</font></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:43_1_1_4"/>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c6">A'</font></span>)</span></span><br/>

<a NAME="E2:43_1_1_4"/><b>then </b>
<font color="Maroon" title="c17">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c6">A'</font>
 
<b>by </b><i><a class="ref" href="vectsp_7.html#T13" title="VECTSP_7:th.13">VECTSP_7:13</a></i>;<br/>
<b>hence </b><a NAME="E3:43_1_1_4"/>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c6">A'</font></span>)</span>
 <b>by </b><i><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</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>
<a NAME="E30:43_1_1"/><b>then </b><i><font color="Green" title="E29">B7</font></i>: 
<font color="Maroon" title="c16">e</font> <a href="aofa_000.html#NR1" title="AOFA_000:NR.1">nin</a> <font color="Maroon" title="c4">A</font>
 
<b>by </b><i><a class="txt" href="#E26:43_1_1"><i><font color="Green" title="E27">A6</font></i></a></i>;<br/>
<a NAME="E31:43_1_1"/><b>then </b><i><font color="Green" title="E30">B8</font></i>: 
<font color="Maroon" title="c16">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B'</font>
 
<b>by </b><i><a class="txt" href="#E27:43_1_1"><i><font color="Green" title="E28">A7</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E32:43_1_1"/>
<font color="Maroon" title="c16">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c4">A</font>
 <b>by </b><i><a class="txt" href="#E30:43_1_1"><i><font color="Green" title="E29">B7</font></i></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c16">e</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <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></span><br/>

<b>reconsider </b><font color="Maroon" title="c17">a</font> = <font color="Maroon" title="c16">e</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">V</font> <b>by </b><i><a class="txt" href="#E31:43_1_1"><i><font color="Green" title="E30">B8</font></i></a></i>;<br/>
<a NAME="E34:43_1_1"/>
<font color="Maroon" title="c6">A'</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c17">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> is <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a>
 
<b>by </b><i><a class="txt" href="#E26:43_1_1"><i><font color="Green" title="E27">A6</font></i></a>, <a class="ref" href="matroid0.html#T13" target="_self" title="MATROID0:th.13">ThI1</a></i>;<br/>
<b>hence </b><a NAME="E35:43_1_1"/>
<font color="Maroon" title="c4">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c16">e</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <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>
 <b>by </b><i><a class="txt" href="#E1:43_1_1"><i><font color="Green" title="E18">A2</font></i></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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