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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="vectsp_1.html#NM2">Field</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="vectsp_1.html#NM6">VectSp</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="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:14"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is <a href="vectsp_7.html#V1">linearly-independent</a>
 <b>and </b><br/><a NAME="E2:14"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="vectsp_9.html#E2:14"><i><font color="Green">E15</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:14_1"/>
not <font color="Maroon">c<sub>5</sub></font> is <a href="vectsp_7.html#V1">linearly-independent</a>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="vectsp_6.html#M2">Linear_Combination</a> of <font color="Maroon">c<sub>5</sub></font><b> such that </b><br/><a NAME="E3:14_1"/><i><font color="Green">E16</font></i>: 
(  <a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">c<sub>6</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="vectsp_7.html#D1">VECTSP_7:def 1</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="vectsp_6.html#M1">Linear_Combination</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E5:14_1"/><i><font color="Green">E17</font></i>: 
(  <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">c<sub>6</sub></font> &amp;  <a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="ref" href="vectsp_9.html#T12" target="_self">Th12</a></i>;<br/><a NAME="E6:14_1"/>
 <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="vectsp_9.html#E5:14_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="vectsp_6.html#D7">VECTSP_6:def 7</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as    <a href="vectsp_6.html#M2">Linear_Combination</a> of <font color="Maroon">c<sub>4</sub></font> <b>by </b><i><a class="ref" href="vectsp_6.html#D7">VECTSP_6:def 7</a></i>;<br/><a NAME="E8:14_1"/>
(  <a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">c<sub>8</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="vectsp_6.html#K1">Carrier</a> <font color="Maroon">c<sub>8</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="vectsp_9.html#E3:14_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="vectsp_9.html#E5:14_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="vectsp_4.html#T19">VECTSP_4:19</a></i>;<br/><b>hence </b><a NAME="E9:14_1"/>
contradiction
 <b>by </b><i><a class="txt" href="vectsp_9.html#E1:14"><i><font color="Green">E14</font></i></a>, <a class="ref" href="vectsp_7.html#D1">VECTSP_7:def 1</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:14"/>
<font color="Maroon">c<sub>4</sub></font> is  <a href="vectsp_7.html#V1">linearly-independent</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>3</sub></font>
 ;<br/>


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