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

<span class="kw">set </span><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/>
<span class="kw">take </span>
<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> 
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> 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> ) )</span><br/>

<a NAME="E1:27_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> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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>  iff  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> ) )</span><br/>

<div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  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> ) implies <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>  )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:27_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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  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> )</span><br/><a NAME="E2:27_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A1</font></span>: 
 ex <font color="Olive" title="b1">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font> st <br/>( <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="Olive" title="b1">A</font> &amp; <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> )
 ;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">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> ;<br/><a NAME="E4:27_1_1_1_1"/>
 <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:27_1_1_1_1"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="ref" href="rusub_4.html#T9" target="_self" title="RUSUB_4:th.9">Th9</a></span>;<br/><span class="kw">hence </span><a NAME="E5:27_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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<span class="kw">given </span><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><span class="kw"> such that </span><a NAME="E3:27_1_1_1"/><span class="lab"><font color="Green" title="E17">A2</font></span>: 
<font color="Maroon" title="c5">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font>
 <span class="kw">and </span><br/><a NAME="E4:27_1_1_1"/><span class="lab"><font color="Green" title="E18">A3</font></span>: 
 <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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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> </span><br/>

<span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E6:27_1_1_1"/><span class="lab"><font color="Green" title="E19">A4</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rusub_4.html#D1" target="_self" title="RUSUB_4:def.1">Def1</a></span>;<br/>
<span class="kw">reconsider </span><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="E8:27_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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:27_1_1_1"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="ref" href="rusub_3.html#D2" title="RUSUB_3:def.2">RUSUB_3:def 2</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="ref" href="rusub_3.html#T22" title="RUSUB_3:th.22">RUSUB_3:22</a></span>;<br/>
<span class="lab"><font color="Green" title="E20">A5</font></span>: <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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:27_1_1_1"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="txt" href="#E6:27_1_1_1"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="ref" href="rusub_3.html#D2" title="RUSUB_3:def.2">RUSUB_3:def 2</a></span>
<br/>.= 
 <a href="rusub_3.html#K1" title="RUSUB_3:func.1">Lin</a> <font color="Maroon" title="c8">B</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rusub_3.html#T28" title="RUSUB_3:th.28">RUSUB_3:28</a></span>
;<br/>
<a NAME="E11:27_1_1_1"/><span class="kw">then </span>
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:27_1_1_1"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="txt" href="#E4:27_1_1_1"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="ref" href="rusub_4.html#T9" target="_self" title="RUSUB_4:th.9">Th9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:27_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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:27_1_1_1"><span class="lab"><font color="Green" title="E20">A5</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:27_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> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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