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<div class="add">

<b>set </b><font color="Maroon" title="c4">S</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Olive" title="b1">A</font></span>)</span> 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> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font> ) </span>}</span> ;<br/>
<a NAME="E1:43_1_1"/><i><font color="Green" title="E33">A1</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Olive" title="b2">A</font></span>)</span> where <font color="Olive" title="b2">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="b2">A</font> is <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Olive" title="b2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font> ) </span>}</span>  iff  ex <font color="Olive" title="b2">W</font> being <a href="vectsp_1.html#V11" title="VECTSP_1:attr.11">strict</a>  <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">V</font> st <br/>( <font color="Olive" title="b2">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> &amp;  <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Olive" title="b2">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font> ) )
 
<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="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:43_1_1_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Olive" title="b1">A</font></span>)</span> 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> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font> ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon" title="c6">A</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">V</font><b> such that </b><br/><a NAME="E3:43_1_1_1_1"/><i><font color="Green" title="E34">A2</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c6">A</font>
 <b>and </b><br/><a NAME="E4:43_1_1_1_1"/><i><font color="Green" title="E35">A3</font></i>: 
<font color="Maroon" title="c6">A</font> is <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a>
 <b>and </b><br/><a NAME="E5:43_1_1_1_1"/><i><font color="Green" title="E36">A4</font></i>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c6">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font>
 ;<br/><b>reconsider </b><font color="Maroon" title="c7">W</font> = <font color="Maroon" title="c5">x</font> as  <a href="vectsp_1.html#V11" title="VECTSP_1:attr.11">strict</a>  <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">V</font> <b>by </b><i><a class="txt" href="vectsp_9.html#E3:43_1_1_1_1"><i><font color="Green" title="E34">A2</font></i></a></i>;<br/><a NAME="E7:43_1_1_1_1"/>
 <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c7">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font>
 <b>by </b><i><a class="txt" href="vectsp_9.html#E3:43_1_1_1_1"><i><font color="Green" title="E34">A2</font></i></a>, <a class="txt" href="vectsp_9.html#E4:43_1_1_1_1"><i><font color="Green" title="E35">A3</font></i></a>, <a class="txt" href="vectsp_9.html#E5:43_1_1_1_1"><i><font color="Green" title="E36">A4</font></i></a>, <a class="ref" href="vectsp_9.html#T30" target="_self" title="VECTSP_9:th.30">Th30</a></i>;<br/><b>hence </b><a NAME="E8:43_1_1_1_1"/>
 ex <font color="Olive" title="b1">W</font> being <a href="vectsp_1.html#V11" title="VECTSP_1:attr.11">strict</a>  <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">V</font> st <br/>( <font color="Olive" title="b1">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> &amp;  <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Olive" title="b1">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font> )
 ;<br/></div>
<b>end;</b></div>

<b>given </b><font color="Maroon" title="c6">W</font> being  <a href="vectsp_1.html#V11" title="VECTSP_1:attr.11">strict</a>  <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">V</font><b> such that </b><a NAME="E2:43_1_1_1"/><i><font color="Green" title="E34">A5</font></i>: 
<font color="Maroon" title="c6">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font>
 <b>and </b><br/><a NAME="E3:43_1_1_1"/><i><font color="Green" title="E35">A6</font></i>: 
 <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c6">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c7">A</font> being  <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 <font color="Maroon" title="c6">W</font><b> such that </b><br/><a NAME="E5:43_1_1_1"/><i><font color="Green" title="E36">A7</font></i>: 
<font color="Maroon" title="c7">A</font> is    <a href="vectsp_7.html#M1" title="VECTSP_7:mode.1">Basis</a> of <font color="Maroon" title="c6">W</font>
 <b>by </b><i><a class="ref" href="vectsp_9.html#D1" target="_self" title="VECTSP_9:def.1">Def1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">A</font> = <font color="Maroon" title="c7">A</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c6">W</font> ;<br/>
<a NAME="E7:43_1_1_1"/>
<font color="Maroon" title="c8">A</font> is <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a>
 
<b>by </b><i><a class="txt" href="vectsp_9.html#E5:43_1_1_1"><i><font color="Green" title="E36">A7</font></i></a>, <a class="ref" href="vectsp_7.html#D3" title="VECTSP_7:def.3">VECTSP_7:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c9">B</font> = <font color="Maroon" title="c8">A</font> as  <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="vectsp_9.html#T15" target="_self" title="VECTSP_9:th.15">Th15</a></i>;<br/>
<i><font color="Green" title="E37">A8</font></i>: <font color="Maroon" title="c5">x</font> = 
 <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c8">A</font>
<b>by </b><i><a class="txt" href="vectsp_9.html#E2:43_1_1_1"><i><font color="Green" title="E34">A5</font></i></a>, <a class="txt" href="vectsp_9.html#E5:43_1_1_1"><i><font color="Green" title="E36">A7</font></i></a>, <a class="ref" href="vectsp_7.html#D3" title="VECTSP_7:def.3">VECTSP_7:def 3</a></i>
<br/>.= 
 <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c9">B</font>
<b>by </b><i><a class="ref" href="vectsp_9.html#T21" target="_self" title="VECTSP_9:th.21">Th21</a></i>
;<br/>
<a NAME="E10:43_1_1_1"/><b>then </b>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c9">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font>
 
<b>by </b><i><a class="txt" href="vectsp_9.html#E2:43_1_1_1"><i><font color="Green" title="E34">A5</font></i></a>, <a class="txt" href="vectsp_9.html#E3:43_1_1_1"><i><font color="Green" title="E35">A6</font></i></a>, <a class="ref" href="vectsp_9.html#T30" target="_self" title="VECTSP_9:th.30">Th30</a></i>;<br/>
<b>hence </b><a NAME="E11:43_1_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Olive" title="b1">A</font></span>)</span> 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> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="vectsp_9.html#E9:43_1_1_1"><i><font color="Green" title="E37">A8</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>take </b>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Olive" title="b1">A</font></span>)</span> 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> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font> ) </span>}</span> 
;<br/>

<b>thus </b><a NAME="E2:43_1_1"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Olive" title="b2">A</font></span>)</span> where <font color="Olive" title="b2">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="b2">A</font> is <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Olive" title="b2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font> ) </span>}</span>  iff  ex <font color="Olive" title="b2">W</font> being <a href="vectsp_1.html#V11" title="VECTSP_1:attr.11">strict</a>  <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">V</font> st <br/>( <font color="Olive" title="b2">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> &amp;  <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Olive" title="b2">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font> ) )
 <b>by </b><i><a class="txt" href="vectsp_9.html#E1:43_1_1"><i><font color="Green" title="E33">A1</font></i></a></i>;<br/>


</div>
