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<b>set </b><font color="Maroon">S</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <font color="Olive">A</font></span>)</span> where <font color="Olive">A</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font> : ( <font color="Olive">A</font> is <a href="vectsp_7.html#V1">linearly-independent</a> &amp;  <a href="card_1.html#K1">Card</a> <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> ) </span>}</span> ;<br/>
<a NAME="E1:43_1_1"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <font color="Olive">A</font></span>)</span> where <font color="Olive">A</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font> : ( <font color="Olive">A</font> is <a href="vectsp_7.html#V1">linearly-independent</a> &amp;  <a href="card_1.html#K1">Card</a> <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> ) </span>}</span>  iff  ex <font color="Olive">W</font> being <a href="vectsp_1.html#V11">strict</a>  <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">V</font> st <br/>( <font color="Olive">W</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> &amp;  <a href="vectsp_9.html#K1" target="_self">dim</a> <font color="Olive">W</font> <a href="hidden.html#R1">=</a> <font color="Maroon">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">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:43_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <font color="Olive">A</font></span>)</span> where <font color="Olive">A</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font> : ( <font color="Olive">A</font> is <a href="vectsp_7.html#V1">linearly-independent</a> &amp;  <a href="card_1.html#K1">Card</a> <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">A</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E3:43_1_1_1_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E4:43_1_1_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">A</font> is <a href="vectsp_7.html#V1">linearly-independent</a>
 <b>and </b><br/><a NAME="E5:43_1_1_1_1"/><i><font color="Green">E43</font></i>: 
 <a href="card_1.html#K1">Card</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 ;<br/><b>reconsider </b><font color="Maroon">W</font> = <font color="Maroon">x</font> as  <a href="vectsp_1.html#V11">strict</a>  <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">V</font> <b>by </b><i/>;<br/><a NAME="E7:43_1_1_1_1"/>
 <a href="vectsp_9.html#K1" target="_self">dim</a> <font color="Maroon">W</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 <b>by </b><i>, , , </i>;<br/><b>hence </b><a NAME="E8:43_1_1_1_1"/>
 ex <font color="Olive">W</font> being <a href="vectsp_1.html#V11">strict</a>  <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">V</font> st <br/>( <font color="Olive">W</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp;  <a href="vectsp_9.html#K1" target="_self">dim</a> <font color="Olive">W</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 ;<br/></div>
<b>end;</b></div>

<b>given </b><font color="Maroon">W</font> being  <a href="vectsp_1.html#V11">strict</a>  <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">V</font><b> such that </b><a NAME="E2:43_1_1_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">W</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E3:43_1_1_1"/><i><font color="Green">E45</font></i>: 
 <a href="vectsp_9.html#K1" target="_self">dim</a> <font color="Maroon">W</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 ;<br/>

<b>consider </b><font color="Maroon">A</font> being  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">W</font><b> such that </b><br/><a NAME="E5:43_1_1_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">A</font> is    <a href="vectsp_7.html#M1">Basis</a> of <font color="Maroon">W</font>
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">A</font> = <font color="Maroon">A</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">W</font> ;<br/>
<a NAME="E7:43_1_1_1"/>
<font color="Maroon">A</font> is <a href="vectsp_7.html#V1">linearly-independent</a>
 
<b>by </b><i>, <a class="ref" href="vectsp_7.html#D3">VECTSP_7:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">B</font> = <font color="Maroon">A</font> as  <a href="vectsp_7.html#V1">linearly-independent</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font> <b>by </b><i/>;<br/>
<i><font color="Green">E47</font></i>: <font color="Maroon">x</font> = 
 <a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">A</font>
<b>by </b><i>, , <a class="ref" href="vectsp_7.html#D3">VECTSP_7:def 3</a></i>
<br/>.= 
 <a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">B</font>
<b>by </b><i/>
;<br/>
<a NAME="E10:43_1_1_1"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E11:43_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <font color="Olive">A</font></span>)</span> where <font color="Olive">A</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font> : ( <font color="Olive">A</font> is <a href="vectsp_7.html#V1">linearly-independent</a> &amp;  <a href="card_1.html#K1">Card</a> <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> ) </span>}</span> 
 <b>by </b><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">Lin</a> <font color="Olive">A</font></span>)</span> where <font color="Olive">A</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font> : ( <font color="Olive">A</font> is <a href="vectsp_7.html#V1">linearly-independent</a> &amp;  <a href="card_1.html#K1">Card</a> <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> ) </span>}</span> 
;<br/>

<b>thus </b><a NAME="E2:43_1_1"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <font color="Olive">A</font></span>)</span> where <font color="Olive">A</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font> : ( <font color="Olive">A</font> is <a href="vectsp_7.html#V1">linearly-independent</a> &amp;  <a href="card_1.html#K1">Card</a> <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> ) </span>}</span>  iff  ex <font color="Olive">W</font> being <a href="vectsp_1.html#V11">strict</a>  <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">V</font> st <br/>( <font color="Olive">W</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> &amp;  <a href="vectsp_9.html#K1" target="_self">dim</a> <font color="Olive">W</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> ) )
 <b>by </b><i><a class="ref" href="vectsp_9.html#T5" target="_self">Th5</a></i>;<br/>


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