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<b>set </b><font color="Maroon" title="c3">S</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="rusub_3.html#K1" title="RUSUB_3: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="c1">V</font> : ( <font color="Olive" title="b1">A</font> is <a href="rlvect_3.html#V1" title="RLVECT_3: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="c2">n</font> ) </span>}</span> ;<br/>
<a NAME="E1:28_1_1"/><i><font color="Green" title="E18">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="rusub_3.html#K1" title="RUSUB_3: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="c1">V</font> : ( <font color="Olive" title="b2">A</font> is <a href="rlvect_3.html#V1" title="RLVECT_3: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="c2">n</font> ) </span>}</span>  iff  ex <font color="Olive" title="b2">W</font> being <a href="bhsp_1.html#V1" title="BHSP_1:attr.1">strict</a>  <a href="rusub_1.html#M1" title="RUSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">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="rusub_4.html#K1" title="RUSUB_4: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="c2">n</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">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="E1:28_1_1_1_1"/>
<font color="Maroon" title="c4">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="rusub_3.html#K1" title="RUSUB_3: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="c1">V</font> : ( <font color="Olive" title="b1">A</font> is <a href="rlvect_3.html#V1" title="RLVECT_3: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="c2">n</font> ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon" title="c5">A</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E3:28_1_1_1_1"/><i><font color="Green" title="E19">A2</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rusub_3.html#K1" title="RUSUB_3:func.1">Lin</a> <font color="Maroon" title="c5">A</font>
 <b>and </b><br/><a NAME="E4:28_1_1_1_1"/><i><font color="Green" title="E20">A3</font></i>: 
<font color="Maroon" title="c5">A</font> is <a href="rlvect_3.html#V1" title="RLVECT_3:attr.1">linearly-independent</a>
 <b>and </b><br/><a NAME="E5:28_1_1_1_1"/><i><font color="Green" title="E21">A4</font></i>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font>
 ;<br/><b>reconsider </b><font color="Maroon" title="c6">W</font> = <font color="Maroon" title="c4">x</font> as  <a href="bhsp_1.html#V1" title="BHSP_1:attr.1">strict</a>  <a href="rusub_1.html#M1" title="RUSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="txt" href="rusub_4.html#E3:28_1_1_1_1"><i><font color="Green" title="E19">A2</font></i></a></i>;<br/><a NAME="E7:28_1_1_1_1"/>
 <a href="rusub_4.html#K1" title="RUSUB_4: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="c2">n</font>
 <b>by </b><i><a class="txt" href="rusub_4.html#E3:28_1_1_1_1"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="rusub_4.html#E4:28_1_1_1_1"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="rusub_4.html#E5:28_1_1_1_1"><i><font color="Green" title="E21">A4</font></i></a>, <a class="ref" href="rusub_4.html#T9" target="_self" title="RUSUB_4:th.9">Th9</a></i>;<br/><b>hence </b><a NAME="E8:28_1_1_1_1"/>
 ex <font color="Olive" title="b1">W</font> being <a href="bhsp_1.html#V1" title="BHSP_1:attr.1">strict</a>  <a href="rusub_1.html#M1" title="RUSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">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="c4">x</font> &amp;  <a href="rusub_4.html#K1" title="RUSUB_4: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="c2">n</font> )
 ;<br/></div>
<b>end;</b></div>

<b>given </b><font color="Maroon" title="c5">W</font> being  <a href="bhsp_1.html#V1" title="BHSP_1:attr.1">strict</a>  <a href="rusub_1.html#M1" title="RUSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font><b> such that </b><a NAME="E2:28_1_1_1"/><i><font color="Green" title="E19">A5</font></i>: 
<font color="Maroon" title="c5">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font>
 <b>and </b><br/><a NAME="E3:28_1_1_1"/><i><font color="Green" title="E20">A6</font></i>: 
 <a href="rusub_4.html#K1" title="RUSUB_4:func.1">dim</a> <font color="Maroon" title="c5">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c6">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="c5">W</font><b> such that </b><br/><a NAME="E5:28_1_1_1"/><i><font color="Green" title="E21">A7</font></i>: 
<font color="Maroon" title="c6">A</font> is    <a href="rusub_3.html#M1" title="RUSUB_3:mode.1">Basis</a> of <font color="Maroon" title="c5">W</font>
 <b>by </b><i><a class="ref" href="rusub_4.html#D2" target="_self" title="RUSUB_4:def.2">Def2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">A</font> = <font color="Maroon" title="c6">A</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c5">W</font> ;<br/>
<a NAME="E7:28_1_1_1"/>
<font color="Maroon" title="c7">A</font> is <a href="rlvect_3.html#V1" title="RLVECT_3:attr.1">linearly-independent</a>
 
<b>by </b><i><a class="txt" href="rusub_4.html#E5:28_1_1_1"><i><font color="Green" title="E21">A7</font></i></a>, <a class="ref" href="rusub_3.html#D2" title="RUSUB_3:def.2">RUSUB_3:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c8">B</font> = <font color="Maroon" title="c7">A</font> as  <a href="rlvect_3.html#V1" title="RLVECT_3:attr.1">linearly-independent</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="ref" href="rusub_3.html#T22" title="RUSUB_3:th.22">RUSUB_3:22</a></i>;<br/>
<i><font color="Green" title="E22">A8</font></i>: <font color="Maroon" title="c4">x</font> = 
 <a href="rusub_3.html#K1" title="RUSUB_3:func.1">Lin</a> <font color="Maroon" title="c7">A</font>
<b>by </b><i><a class="txt" href="rusub_4.html#E2:28_1_1_1"><i><font color="Green" title="E19">A5</font></i></a>, <a class="txt" href="rusub_4.html#E5:28_1_1_1"><i><font color="Green" title="E21">A7</font></i></a>, <a class="ref" href="rusub_3.html#D2" title="RUSUB_3:def.2">RUSUB_3:def 2</a></i>
<br/>.= 
 <a href="rusub_3.html#K1" title="RUSUB_3:func.1">Lin</a> <font color="Maroon" title="c8">B</font>
<b>by </b><i><a class="ref" href="rusub_3.html#T28" title="RUSUB_3:th.28">RUSUB_3:28</a></i>
;<br/>
<a NAME="E10:28_1_1_1"/><b>then </b>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c8">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font>
 
<b>by </b><i><a class="txt" href="rusub_4.html#E2:28_1_1_1"><i><font color="Green" title="E19">A5</font></i></a>, <a class="txt" href="rusub_4.html#E3:28_1_1_1"><i><font color="Green" title="E20">A6</font></i></a>, <a class="ref" href="rusub_4.html#T9" target="_self" title="RUSUB_4:th.9">Th9</a></i>;<br/>
<b>hence </b><a NAME="E11:28_1_1_1"/>
<font color="Maroon" title="c4">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="rusub_3.html#K1" title="RUSUB_3: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="c1">V</font> : ( <font color="Olive" title="b1">A</font> is <a href="rlvect_3.html#V1" title="RLVECT_3: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="c2">n</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="rusub_4.html#E9:28_1_1_1"><i><font color="Green" title="E22">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="rusub_3.html#K1" title="RUSUB_3: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="c1">V</font> : ( <font color="Olive" title="b1">A</font> is <a href="rlvect_3.html#V1" title="RLVECT_3: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="c2">n</font> ) </span>}</span> 
;<br/>

<b>thus </b><a NAME="E2:28_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="rusub_3.html#K1" title="RUSUB_3: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="c1">V</font> : ( <font color="Olive" title="b2">A</font> is <a href="rlvect_3.html#V1" title="RLVECT_3: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="c2">n</font> ) </span>}</span>  iff  ex <font color="Olive" title="b2">W</font> being <a href="bhsp_1.html#V1" title="BHSP_1:attr.1">strict</a>  <a href="rusub_1.html#M1" title="RUSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">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="rusub_4.html#K1" title="RUSUB_4: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="c2">n</font> ) )
 <b>by </b><i><a class="txt" href="rusub_4.html#E1:28_1_1"><i><font color="Green" title="E18">A1</font></i></a></i>;<br/>


</div>
