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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>;<br/>

<b>reconsider </b><font color="Maroon" title="c2">V'</font> =  <a href="rusub_1.html#K1" title="RUSUB_1:func.1">(0).</a> <font color="Maroon" title="c1">V</font> as  <a href="bhsp_1.html#V1" title="BHSP_1:attr.1">strict</a> <a href="bhsp_1.html#NM1" title="BHSP_1:NM.1">RealUnitarySpace</a> ;<br/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V'</font> = 
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">V</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
<b>by </b><i><a class="ref" href="rusub_1.html#D2" title="RUSUB_1:def.2">RUSUB_1:def 2</a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c2">V'</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
<b>by </b><i><a class="ref" href="rusub_1.html#T4" title="RUSUB_1:th.4">RUSUB_1:4</a></i>
<br/>.= 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="rusub_1.html#K1" title="RUSUB_1:func.1">(0).</a> <font color="Maroon" title="c2">V'</font></span>)</span>
<b>by </b><i><a class="ref" href="rusub_1.html#D2" title="RUSUB_1:def.2">RUSUB_1:def 2</a></i>
;<br/>
<a NAME="E3:11"/><b>then </b><i><font color="Green" title="E10">A1</font></i>: 
<font color="Maroon" title="c2">V'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rusub_1.html#K1" title="RUSUB_1:func.1">(0).</a> <font color="Maroon" title="c2">V'</font>
 
<b>by </b><i><a class="ref" href="rusub_1.html#T26" title="RUSUB_1:th.26">RUSUB_1:26</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c3">I</font> =  <a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V'</font> as  <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="c2">V'</font> ;<br/>
<a NAME="E5:11"/><i><font color="Green" title="E11">A2</font></i>: 
<font color="Maroon" title="c3">I</font> is <a href="rlvect_3.html#V1" title="RLVECT_3:attr.1">linearly-independent</a>
 
<b>by </b><i><a class="ref" href="rlvect_3.html#T8" title="RLVECT_3:th.8">RLVECT_3:8</a></i>;<br/>
<a NAME="E6:11"/>
 <a href="rusub_3.html#K1" title="RUSUB_3:func.1">Lin</a> <font color="Maroon" title="c3">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rusub_1.html#K1" title="RUSUB_1:func.1">(0).</a> <font color="Maroon" title="c2">V'</font>
 
<b>by </b><i><a class="ref" href="rusub_3.html#T3" title="RUSUB_3:th.3">RUSUB_3:3</a></i>;<br/>
<a NAME="E7:11"/><b>then </b>
<font color="Maroon" title="c3">I</font> is    <a href="rusub_3.html#M1" title="RUSUB_3:mode.1">Basis</a> of <font color="Maroon" title="c2">V'</font>
 
<b>by </b><i><a class="txt" href="rusub_4.html#E3:11"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="rusub_4.html#E5:11"><i><font color="Green" title="E11">A2</font></i></a>, <a class="ref" href="rusub_3.html#D2" title="RUSUB_3:def.2">RUSUB_3:def 2</a></i>;<br/>
<b>hence </b><a NAME="E8:11"/>
 <a href="rusub_1.html#K1" title="RUSUB_1:func.1">(0).</a> <font color="Maroon" title="c1">V</font> is <a href="rusub_4.html#V1" title="RUSUB_4:attr.1">finite-dimensional</a>
 <b>by </b><i><a class="ref" href="rusub_4.html#D2" target="_self" title="RUSUB_4:def.2">Def2</a></i>;<br/>


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