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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="rlvect_1.html#NM2">RealLinearSpace</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="rlsub_1.html#M1">Subspace</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:16"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="rlvect_3.html#V1">linearly-independent</a>
 <b>and </b><br/><a NAME="E2:16"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="rlvect_5.html#E2:16"><i><font color="Green">E18</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:16_1"/>
not <font color="Maroon">c<sub>4</sub></font> is <a href="rlvect_3.html#V1">linearly-independent</a>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="rlvect_2.html#M2">Linear_Combination</a> of <font color="Maroon">c<sub>4</sub></font><b> such that </b><br/><a NAME="E3:16_1"/><i><font color="Green">E19</font></i>: 
(  <a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="ref" href="rlvect_3.html#D1">RLVECT_3:def 1</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="rlvect_2.html#M1">Linear_Combination</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E5:16_1"/><i><font color="Green">E20</font></i>: 
(  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>5</sub></font> &amp;  <a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="rlvect_5.html#T12" target="_self">Th12</a></i>;<br/><a NAME="E6:16_1"/>
 <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="rlvect_5.html#E5:16_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="rlvect_2.html#D8">RLVECT_2:def 8</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as    <a href="rlvect_2.html#M2">Linear_Combination</a> of <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="ref" href="rlvect_2.html#D8">RLVECT_2:def 8</a></i>;<br/><a NAME="E8:16_1"/>
(  <a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="txt" href="rlvect_5.html#E3:16_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="rlvect_5.html#E5:16_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="rlsub_1.html#T19">RLSUB_1:19</a></i>;<br/><b>hence </b><a NAME="E9:16_1"/>
contradiction
 <b>by </b><i><a class="txt" href="rlvect_5.html#E1:16"><i><font color="Green">E17</font></i></a>, <a class="ref" href="rlvect_3.html#D1">RLVECT_3:def 1</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:16"/>
<font color="Maroon">c<sub>3</sub></font> is  <a href="rlvect_3.html#V1">linearly-independent</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font>
 ;<br/>


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