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<b>let </b><font color="Maroon">V</font> be   <a href="bhsp_1.html#NM1">RealUnitarySpace</a>;<br/><b>let </b><font color="Maroon">W</font> be    <a href="rusub_1.html#M1">Subspace</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">L</font> be    <a href="rlvect_2.html#M1">Linear_Combination</a> of <font color="Maroon">V</font>;<br/>





<b>assume </b><a NAME="E1:27"/><i><font color="Green">E24</font></i>: 
 <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">C</font> =  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L</font> as  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">W</font> ;<br/>
<a NAME="E3:27"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>
 
<b>by </b><i><a class="ref" href="rusub_1.html#D1">RUSUB_1:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">K</font> = <font color="Maroon">L</font> <a href="relat_1.html#K7">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W</font> as   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W</font>, <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="funct_2.html#T38">FUNCT_2:38</a></i>;<br/>
<a NAME="E5:27"/><i><font color="Green">E28</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">K</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<div><i><font color="Green">E29</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">w</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">W</font>;<br/><b>assume </b><a NAME="E1:27_1"/>
not <font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 ;<br/><a NAME="E2:27_1"/><b>then </b>
( not <font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; <font color="Maroon">w</font> is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> )
 <b>by </b><i><a class="ref" href="rusub_1.html#T3">RUSUB_1:3</a></i>;<br/><a NAME="E3:27_1"/><b>then </b>
<font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><b>hence </b><a NAME="E4:27_1"/>
<font color="Maroon">K</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/></div><b>end;</b></div>
<a NAME="E7:27"/>
<font color="Maroon">K</font> is    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W</font>,<a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#T11">FUNCT_2:11</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">K</font> = <font color="Maroon">K</font> as    <a href="rlvect_2.html#M1">Linear_Combination</a> of <font color="Maroon">W</font> <b>by </b><i>, <a class="ref" href="rlvect_2.html#D5">RLVECT_2:def 5</a></i>;<br/>
<b>take </b>
<font color="Maroon">K</font>
;<br/>

<b>thus </b><a NAME="E9:27"/>
(  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">K</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L</font> &amp;  <a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">K</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">L</font> )
 <b>by </b><i>, </i>;<br/>


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