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<b>let </b><font color="Maroon" title="c1">V</font> be  <a href="rusub_4.html#V1" title="RUSUB_4:attr.1">finite-dimensional</a> <a href="bhsp_1.html#NM1" title="BHSP_1:NM.1">RealUnitarySpace</a>;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:21_1"/><i><font color="Green" title="E16">A1</font></i>: 
 <a href="rusub_4.html#K1" title="RUSUB_4:func.1">dim</a> <font color="Maroon" title="c1">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><b>consider </b><font color="Maroon" title="c2">I</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="c1">V</font><b> such that </b><br/><a NAME="E3:21_1"/><i><font color="Green" title="E17">A2</font></i>: 
<font color="Maroon" title="c2">I</font> is    <a href="rusub_3.html#M1" title="RUSUB_3:mode.1">Basis</a> of <font color="Maroon" title="c1">V</font>
 <b>by </b><i><a class="ref" href="rusub_4.html#D2" target="_self" title="RUSUB_4:def.2">Def2</a></i>;<br/><a NAME="E4:21_1"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c2">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="txt" href="rusub_4.html#E1:21_1"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="rusub_4.html#E3:21_1"><i><font color="Green" title="E17">A2</font></i></a>, <a class="ref" href="rusub_4.html#D3" target="_self" title="RUSUB_4:def.3">Def3</a></i>;<br/><a NAME="E5:21_1"/><b>then </b><i><font color="Green" title="E18">A3</font></i>: 
<font color="Maroon" title="c2">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c1">V</font>
 <b>by </b><i><a class="ref" href="card_2.html#T59" title="CARD_2:th.59">CARD_2:59</a></i>;<br/> <a href="rusub_1.html#K2" title="RUSUB_1:func.2">(Omega).</a> <font color="Maroon" title="c1">V</font> = 
 <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>,the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">U2</a> of <font color="Maroon" title="c1">V</font>,the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c1">V</font>,the <a href="rlvect_1.html#U2" title="RLVECT_1:sel.2">Mult</a> of <font color="Maroon" title="c1">V</font>,the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c1">V</font> #)
<b>by </b><i><a class="ref" href="rusub_1.html#D3" title="RUSUB_1:def.3">RUSUB_1:def 3</a></i>
<br/>.= 
 <a href="rusub_3.html#K1" title="RUSUB_3:func.1">Lin</a> <font color="Maroon" title="c2">I</font>
<b>by </b><i><a class="txt" href="rusub_4.html#E3:21_1"><i><font color="Green" title="E17">A2</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_1.html#K1" title="RUSUB_1:func.1">(0).</a> <font color="Maroon" title="c1">V</font>
<b>by </b><i><a class="txt" href="rusub_4.html#E5:21_1"><i><font color="Green" title="E18">A3</font></i></a>, <a class="ref" href="rusub_3.html#T3" title="RUSUB_3:th.3">RUSUB_3:3</a></i>
;<br/><b>hence </b><a NAME="E7:21_1"/>
 <a href="rusub_1.html#K2" title="RUSUB_1:func.2">(Omega).</a> <font color="Maroon" title="c1">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="c1">V</font>
 ;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:21"/>
 <a href="rusub_1.html#K2" title="RUSUB_1:func.2">(Omega).</a> <font color="Maroon" title="c1">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="c1">V</font>
 ;<br/>

<a NAME="E3:21"/><b>then </b><i><font color="Green" title="E16">A4</font></i>: 
 <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>,the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">U2</a> of <font color="Maroon" title="c1">V</font>,the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c1">V</font>,the <a href="rlvect_1.html#U2" title="RLVECT_1:sel.2">Mult</a> of <font color="Maroon" title="c1">V</font>,the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c1">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="c1">V</font>
 
<b>by </b><i><a class="ref" href="rusub_1.html#D3" title="RUSUB_1:def.3">RUSUB_1:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c2">I</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="c1">V</font><b> such that </b><br/><a NAME="E5:21"/><i><font color="Green" title="E17">A5</font></i>: 
<font color="Maroon" title="c2">I</font> is    <a href="rusub_3.html#M1" title="RUSUB_3:mode.1">Basis</a> of <font color="Maroon" title="c1">V</font>
 <b>by </b><i><a class="ref" href="rusub_4.html#D2" target="_self" title="RUSUB_4:def.2">Def2</a></i>;<br/>
<a NAME="E6:21"/>
 <a href="rusub_3.html#K1" title="RUSUB_3:func.1">Lin</a> <font color="Maroon" title="c2">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="c1">V</font>
 
<b>by </b><i><a class="txt" href="rusub_4.html#E3:21"><i><font color="Green" title="E16">A4</font></i></a>, <a class="txt" href="rusub_4.html#E5:21"><i><font color="Green" title="E17">A5</font></i></a>, <a class="ref" href="rusub_3.html#D2" title="RUSUB_3:def.2">RUSUB_3:def 2</a></i>;<br/>
<a NAME="E7:21"/><b>then </b><i><font color="Green" title="E18">A6</font></i>: 
( <font color="Maroon" title="c2">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c2">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="struct_0.html#K4" title="STRUCT_0:func.4">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_3.html#T4" title="RUSUB_3:th.4">RUSUB_3:4</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:21_2"/>
<font color="Maroon" title="c2">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="struct_0.html#K4" title="STRUCT_0:func.4">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>
 ;<br/><a NAME="E2:21_2"/><b>then </b>
not <font color="Maroon" title="c2">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#T9" title="RLVECT_3:th.9">RLVECT_3:9</a></i>;<br/><b>hence </b><a NAME="E3:21_2"/>
contradiction
 <b>by </b><i><a class="txt" href="rusub_4.html#E5:21"><i><font color="Green" title="E17">A5</font></i></a>, <a class="ref" href="rusub_3.html#D2" title="RUSUB_3:def.2">RUSUB_3:def 2</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E9:21"/>
 <a href="rusub_4.html#K1" title="RUSUB_4:func.1">dim</a> <font color="Maroon" title="c1">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="txt" href="rusub_4.html#E5:21"><i><font color="Green" title="E17">A5</font></i></a>, <a class="txt" href="rusub_4.html#E7:21"><i><font color="Green" title="E18">A6</font></i></a>, <a class="ref" href="rusub_4.html#D3" target="_self" title="RUSUB_4:def.3">Def3</a>, <a class="ref" href="card_1.html#T47" title="CARD_1:th.47">CARD_1:47</a></i>;<br/>


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