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<b>let </b><font color="Maroon">V</font> be   <a href="rlvect_1.html#NM2">RealLinearSpace</a>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">L</font> be    <a href="rlvect_2.html#M1" target="_self">Linear_Combination</a> of <font color="Maroon">V</font>;<br/>







<b>assume </b><a NAME="E1:57"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">L</font> is    <a href="rlvect_2.html#M2" target="_self">Linear_Combination</a> of <font color="Maroon">A</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:57_1_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_1_1_1"><b>suppose </b></a><a NAME="E1:57_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:57_1_1_1"/><b>then </b>
<font color="Maroon">a</font> <a href="rlvect_2.html#K8" target="_self">*</a> <font color="Maroon">L</font> <a href="rlvect_2.html#R1" target="_self">=</a>  <a href="rlvect_2.html#K4" target="_self">ZeroLC</a> <font color="Maroon">V</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:57_1_1_1"/>
<font color="Maroon">a</font> <a href="rlvect_2.html#K8" target="_self">*</a> <font color="Maroon">L</font> is    <a href="rlvect_2.html#M2" target="_self">Linear_Combination</a> of <font color="Maroon">A</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_1_1_2"><b>suppose </b></a><a NAME="E1:57_1_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:57_1_1_2"/><b>then </b>
(  <a href="rlvect_2.html#K3" target="_self">Carrier</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_2.html#K8" target="_self">*</a> <font color="Maroon">L</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_2.html#K3" target="_self">Carrier</a> <font color="Maroon">L</font> &amp;  <a href="rlvect_2.html#K3" target="_self">Carrier</a> <font color="Maroon">L</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> )
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E3:57_1_1_2"/>
<font color="Maroon">a</font> <a href="rlvect_2.html#K8" target="_self">*</a> <font color="Maroon">L</font> is    <a href="rlvect_2.html#M2" target="_self">Linear_Combination</a> of <font color="Maroon">A</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:57"/>
<font color="Maroon">a</font> <a href="rlvect_2.html#K8" target="_self">*</a> <font color="Maroon">L</font> is    <a href="rlvect_2.html#M2" target="_self">Linear_Combination</a> of <font color="Maroon">A</font>
 ;<br/>


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