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

<span class="kw">set </span><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"/><span class="lab"><font color="Green" title="E18">A1</font></span>: 
 <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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matroid0.html#D9" target="_self" title="MATROID0:def.9">Def9</a></span>;<br/>
<span class="kw">let </span><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>; <span class="comment"><font color="firebrick">:: according to </font></span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:43_1_1"/><span class="lab"><font color="Green" title="E19">A2</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E3:43_1_1"/><span class="lab"><font color="Green" title="E20">A3</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E4:43_1_1"/><span class="lab"><font color="Green" title="E21">A4</font></span>: 
 <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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="E5:43_1_1"/><span class="lab"><font color="Green" title="E22">A5</font></span>: 
<font color="Maroon" title="c5">B</font> is  <a href="matroid0.html#NV1" title="MATROID0:NV.1">independent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_1_1"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="ref" href="matroid0.html#D2" target="_self" title="MATROID0:def.2">Def2</a></span>;<br/>
<a NAME="E6:43_1_1"/>
<font color="Maroon" title="c4">A</font> is  <a href="matroid0.html#NV1" title="MATROID0:NV.1">independent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_1_1"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="ref" href="matroid0.html#D2" target="_self" title="MATROID0:def.2">Def2</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">A9</font> = <font color="Maroon" title="c4">A</font>, <font color="Maroon" title="c7">B9</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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:43_1_1"><span class="lab"><font color="Green" title="E22">A5</font></span></a>, <a class="ref" href="matroid0.html#T11" target="_self" title="MATROID0:th.11">Th11</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">V9</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">A9</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">B9</font></span>)</span>;<br/>
<a NAME="E8:43_1_1"/>
<font color="Maroon" title="c6">A9</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">A9</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">B9</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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">A9</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">A9</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">B9</font></span>)</span></span>)</span> )</span><br/>

<span class="kw">assume </span><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">A9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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">A9</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">B9</font></span>)</span></span>)</span></span><br/>

<a NAME="E2:43_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">A9</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">B9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E3:43_1_1_1"/><span class="kw">then </span>
<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">A9</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">B9</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_7.html#T8" title="VECTSP_7:th.8">VECTSP_7:8</a></span>;<br/>
<span class="kw">hence </span><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">A9</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">B9</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">A99</font> = <font color="Maroon" title="c6">A9</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">A9</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">B9</font></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_9.html#T12" title="VECTSP_9:th.12">VECTSP_9:12</a></span>;<br/>
<a NAME="E10:43_1_1"/>
<font color="Maroon" title="c7">B9</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">A9</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">B9</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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">B9</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">A9</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">B9</font></span>)</span></span>)</span> )</span><br/>

<span class="kw">assume </span><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">B9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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">A9</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">B9</font></span>)</span></span>)</span></span><br/>

<a NAME="E2:43_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">A9</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">B9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E3:43_1_1_2"/><span class="kw">then </span>
<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">A9</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">B9</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_7.html#T8" title="VECTSP_7:th.8">VECTSP_7:8</a></span>;<br/>
<span class="kw">hence </span><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">A9</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">B9</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">B99</font> = <font color="Maroon" title="c7">B9</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">A9</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">B9</font></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_9.html#T12" title="VECTSP_9:th.12">VECTSP_9:12</a></span>;<br/>
<a NAME="E12:43_1_1"/><span class="lab"><font color="Green" title="E23">A6</font></span>: 
 <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">A9</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">B9</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">A99</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c10">B99</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_9.html#T17" title="VECTSP_9:th.17">VECTSP_9:17</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><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">A9</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">B9</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E14:43_1_1"/><span class="lab"><font color="Green" title="E24">A7</font></span>: 
<font color="Maroon" title="c7">B9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">D</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_7.html#T19" title="VECTSP_7:th.19">VECTSP_7:19</a></span>;<br/>
<span class="kw">consider </span><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">A9</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">B9</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E16:43_1_1"/><span class="lab"><font color="Green" title="E25">A8</font></span>: 
<font color="Maroon" title="c12">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">A99</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c10">B99</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:43_1_1"><span class="lab"><font color="Green" title="E23">A6</font></span></a>, <a class="ref" href="vectsp_7.html#T20" title="VECTSP_7:th.20">VECTSP_7:20</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:43_1_1"><span class="lab"><font color="Green" title="E25">A8</font></span></a></span>;<br/>
<a NAME="E18:43_1_1"/>
<font color="Maroon" title="c13">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">A9</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">B9</font></span>)</span>
 
;<br/>
<a NAME="E19:43_1_1"/><span class="kw">then </span>
 <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">A9</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">B9</font></span>)</span> is  <a href="matrlin.html#V1" title="MATRLIN:attr.1">finite-dimensional</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrlin.html#D1" title="MATRLIN:def.1">MATRLIN:def 1</a></span>;<br/>
<a NAME="E20:43_1_1"/><span class="kw">then </span>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c13">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_9.html#T22" title="VECTSP_9:th.22">VECTSP_9:22</a></span>;<br/>
<a NAME="E21:43_1_1"/><span class="kw">then </span>
 <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="c13">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:43_1_1"><span class="lab"><font color="Green" title="E24">A7</font></span></a>, <a class="ref" href="card_1.html#T11" title="CARD_1:th.11">CARD_1:11</a></span>;<br/>
<a NAME="E22:43_1_1"/><span class="kw">then </span>
 <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="c13">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T39" title="NAT_1:th.39">NAT_1:39</a></span>;<br/>
<a NAME="E23:43_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A9</font></span>: 
 <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="c13">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43_1_1"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c14">e</font> =  the    <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">A9</font></span>)</span>;<br/>
<a NAME="E24:43_1_1"/><span class="lab"><font color="Green" title="E27">A10</font></span>: 
<font color="Maroon" title="c6">A9</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">A9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ranknull.html#T20" title="RANKNULL:th.20">RANKNULL:20</a></span>;<br/>
<a NAME="E25:43_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A11</font></span>: 
 <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c6">A9</font> is  <a href="matrlin.html#V1" title="MATRLIN:attr.1">finite-dimensional</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrlin.html#D1" title="MATRLIN:def.1">MATRLIN:def 1</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1_3"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><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">A9</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c15">C9</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">A9</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">A9</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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_4.html#D2" title="VECTSP_4:def.2">VECTSP_4:def 2</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c16">C99</font> = <font color="Maroon" title="c15">C9</font> as   <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="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_7.html#D3" title="VECTSP_7:def.3">VECTSP_7:def 3</a></span>;<br/><a NAME="E6:43_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c16">C99</font> is  <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_9.html#T11" title="VECTSP_9:th.11">VECTSP_9:11</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><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">A9</font><span class="kw"> such that </span><br/><a NAME="E8:43_1_1_3"/><span class="lab"><font color="Green" title="E29">A12</font></span>: 
<font color="Maroon" title="c15">C9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c17">E</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_7.html#T19" title="VECTSP_7:th.19">VECTSP_7:19</a>, <a class="ref" href="vectsp_9.html#T12" title="VECTSP_9:th.12">VECTSP_9:12</a></span>;<br/><a NAME="E9:43_1_1_3"/><span class="lab"><font color="Green" title="E30">A13</font></span>: 
 <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:43_1_1"><span class="lab"><font color="Green" title="E27">A10</font></span></a>, <a class="txt" href="#E25:43_1_1"><span class="lab"><font color="Green" title="E28">A11</font></span></a>, <a class="ref" href="vectsp_9.html#T22" title="VECTSP_9:th.22">VECTSP_9:22</a></span>;<br/><a NAME="E10:43_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c17">E</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 ;<br/><span class="kw">hence </span><a NAME="E11:43_1_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:43_1_1"><span class="lab"><font color="Green" title="E26">A9</font></span></a>, <a class="txt" href="#E8:43_1_1_3"><span class="lab"><font color="Green" title="E29">A12</font></span></a>, <a class="txt" href="#E9:43_1_1_3"><span class="lab"><font color="Green" title="E30">A13</font></span></a>, <a class="ref" href="nat_1.html#T43" title="NAT_1:th.43">NAT_1:43</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E28:43_1_1"/><span class="lab"><font color="Green" title="E29">A14</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E29:43_1_1"/><span class="lab"><font color="Green" title="E30">A15</font></span>: 
<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">A9</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/>
<a NAME="E30:43_1_1"/><span class="lab"><font color="Green" title="E31">A16</font></span>: 
<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">A9</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:43_1_1"><span class="lab"><font color="Green" title="E29">A14</font></span></a>, <a class="txt" href="#E29:43_1_1"><span class="lab"><font color="Green" title="E30">A15</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E31:43_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A17</font></span>: 
 the    <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">A9</font></span>)</span> <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">A9</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E32:43_1_1"/><span class="lab"><font color="Green" title="E33">A18</font></span>: 
 the    <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">A9</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:43_1_1"><span class="lab"><font color="Green" title="E31">A16</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E33:43_1_1"/><span class="kw">then </span>
 the    <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">A9</font></span>)</span> <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:43_1_1"><span class="lab"><font color="Green" title="E25">A8</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c16">e</font> =  the    <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">A9</font></span>)</span> 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/>
<span class="kw">take </span>
<font color="Maroon" title="c16">e</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="E35:43_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">A9</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c17">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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">A9</font></span>)</span> )</span><br/>

<span class="kw">assume </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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">A9</font></span>)</span></span><br/>

<a NAME="E2:43_1_1_4"/><span class="kw">then </span>
<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">A9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_7.html#T8" title="VECTSP_7:th.8">VECTSP_7:8</a></span>;<br/>
<span class="kw">hence </span><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">A9</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E36:43_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A19</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:43_1_1"><span class="lab"><font color="Green" title="E31">A16</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E37:43_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A20</font></span>: 
<font color="Maroon" title="c16">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:43_1_1"><span class="lab"><font color="Green" title="E25">A8</font></span></a>, <a class="txt" href="#E32:43_1_1"><span class="lab"><font color="Green" title="E33">A18</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E38:43_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:43_1_1"><span class="lab"><font color="Green" title="E34">A19</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:43_1_1"><span class="lab"><font color="Green" title="E35">A20</font></span></a></span>;<br/>
<a NAME="E40:43_1_1"/>
<font color="Maroon" title="c6">A9</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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:43_1_1"><span class="lab"><font color="Green" title="E32">A17</font></span></a>, <a class="ref" href="matroid0.html#T13" target="_self" title="MATROID0:th.13">Th13</a></span>;<br/>
<span class="kw">hence </span><a NAME="E41:43_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_1_1"><span class="lab"><font color="Green" title="E18">A1</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
