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<b>let </b><font color="Maroon" title="c1">V</font> be   <a href="bhsp_1.html#NM1" title="BHSP_1:NM.1">RealUnitarySpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">W</font> being   <a href="rusub_1.html#M1" title="RUSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font><br/> for <font color="Olive" title="b2">A</font> being   <a href="rusub_3.html#M1" title="RUSUB_3:mode.1">Basis</a> of <font color="Olive" title="b1">W</font>  ex <font color="Olive" title="b3">B</font> being   <a href="rusub_3.html#M1" title="RUSUB_3:mode.1">Basis</a> of <font color="Maroon" title="c1">V</font> st <font color="Olive" title="b2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">B</font></span><br/><b>let </b><font color="Maroon" title="c2">W</font> be    <a href="rusub_1.html#M1" title="RUSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">A</font> being   <a href="rusub_3.html#M1" title="RUSUB_3:mode.1">Basis</a> of <font color="Maroon" title="c2">W</font>  ex <font color="Olive" title="b2">B</font> being   <a href="rusub_3.html#M1" title="RUSUB_3:mode.1">Basis</a> of <font color="Maroon" title="c1">V</font> st <font color="Olive" title="b1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">B</font></span><br/><b>let </b><font color="Maroon" title="c3">A</font> be    <a href="rusub_3.html#M1" title="RUSUB_3:mode.1">Basis</a> of <font color="Maroon" title="c2">W</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">B</font> being   <a href="rusub_3.html#M1" title="RUSUB_3:mode.1">Basis</a> of <font color="Maroon" title="c1">V</font> st <font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">B</font></span><br/>





<a NAME="E1:31"/>
( <font color="Maroon" title="c3">A</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">W</font> &amp; <font color="Maroon" title="c3">A</font> is <a href="rlvect_3.html#V1" title="RLVECT_3:attr.1">linearly-independent</a> )
 
<b>by </b><i><a class="ref" href="rusub_3.html#D2" target="_self" title="RUSUB_3:def.2">Def2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c4">B</font> = <font color="Maroon" title="c3">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" target="_self" title="RUSUB_3:th.22">Th22</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">I</font> being    <a href="rusub_3.html#M1" title="RUSUB_3:mode.1">Basis</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E4:31"/><i><font color="Green" title="E21">A1</font></i>: 
<font color="Maroon" title="c4">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">I</font>
 <b>by </b><i><a class="ref" href="rusub_3.html#T15" target="_self" title="RUSUB_3:th.15">Th15</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c5">I</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">I</font></span><br/>

<b>thus </b><a NAME="E5:31"/>
<font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">I</font>
 <b>by </b><i><a class="txt" href="#E4:31"><i><font color="Green" title="E21">A1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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