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<b>let </b><font color="Maroon" title="c5">F1</font>, <font color="Maroon" title="c6">F2</font> be   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">K</font>;<br/>

<b>assume </b><a NAME="E1:51_1_2"/><i><font color="Green" title="E39">A6</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">b1</font>
 ;<br/>

<b>given </b><font color="Maroon" title="c7">KL1</font> being    <a href="vectsp_6.html#M1" title="VECTSP_6:mode.1">Linear_Combination</a> of <font color="Maroon" title="c2">V</font><b> such that </b><a NAME="E3:51_1_2"/><i><font color="Green" title="E40">A7</font></i>: 
<font color="Maroon" title="c4">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c7">KL1</font>
 <b>and </b><br/><a NAME="E4:51_1_2"/><i><font color="Green" title="E41">A8</font></i>: 
 <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c7">KL1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">b1</font>
 <b>and </b><br/><a NAME="E5:51_1_2"/><i><font color="Green" title="E42">A9</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F1</font> holds <br/><font color="Maroon" title="c5">F1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">KL1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">k</font></span>)</span>
 ;<br/>

<b>assume </b><a NAME="E6:51_1_2"/><i><font color="Green" title="E43">A10</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">F2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">b1</font>
 ;<br/>

<b>given </b><font color="Maroon" title="c8">KL2</font> being    <a href="vectsp_6.html#M1" title="VECTSP_6:mode.1">Linear_Combination</a> of <font color="Maroon" title="c2">V</font><b> such that </b><a NAME="E8:51_1_2"/><i><font color="Green" title="E44">A11</font></i>: 
<font color="Maroon" title="c4">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c8">KL2</font>
 <b>and </b><br/><a NAME="E9:51_1_2"/><i><font color="Green" title="E45">A12</font></i>: 
 <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c8">KL2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">b1</font>
 <b>and </b><br/><a NAME="E10:51_1_2"/><i><font color="Green" title="E46">A13</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">F2</font> holds <br/><font color="Maroon" title="c6">F2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">KL2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">k</font></span>)</span>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c9">b</font> =  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">b1</font> as    <a href="vectsp_7.html#M1" title="VECTSP_7:mode.1">Basis</a> of <font color="Maroon" title="c2">V</font> <b>by </b><i><a class="ref" href="matrlin.html#D4" target="_self" title="MATRLIN:def.4">Def4</a></i>;<br/>
<a NAME="E12:51_1_2"/><i><font color="Green" title="E47">A14</font></i>: 
<font color="Maroon" title="c9">b</font> is <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a>
 
<b>by </b><i><a class="ref" href="vectsp_7.html#D3" title="VECTSP_7:def.3">VECTSP_7:def 3</a></i>;<br/>
<a NAME="E13:51_1_2"/>
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F1</font> holds <br/><font color="Maroon" title="c5">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:51_1_2_1"/><i><font color="Green" title="E48">A15</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k</font> &amp; <font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F1</font> )
 ;<br/>

<a NAME="E2:51_1_2_1"/><b>then </b><i><font color="Green" title="E49">A16</font></i>: 
<font color="Maroon" title="c10">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c5">F1</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E3:51_1_2_1"/><i><font color="Green" title="E50">A17</font></i>: 
<font color="Maroon" title="c10">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c6">F2</font>
 
<b>by </b><i><a class="txt" href="matrlin.html#E1:51_1_2"><i><font color="Green" title="E39">A6</font></i></a>, <a class="txt" href="matrlin.html#E6:51_1_2"><i><font color="Green" title="E43">A10</font></i></a>, <a class="txt" href="matrlin.html#E1:51_1_2_1"><i><font color="Green" title="E48">A15</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c5">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> = 
<font color="Maroon" title="c5">F1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c10">k</font>
<b>by </b><i><a class="txt" href="matrlin.html#E2:51_1_2_1"><i><font color="Green" title="E49">A16</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon" title="c7">KL1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c10">k</font></span>)</span>
<b>by </b><i><a class="txt" href="matrlin.html#E5:51_1_2"><i><font color="Green" title="E42">A9</font></i></a>, <a class="txt" href="matrlin.html#E1:51_1_2_1"><i><font color="Green" title="E48">A15</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c8">KL2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c10">k</font></span>)</span>
<b>by </b><i><a class="txt" href="matrlin.html#E3:51_1_2"><i><font color="Green" title="E40">A7</font></i></a>, <a class="txt" href="matrlin.html#E4:51_1_2"><i><font color="Green" title="E41">A8</font></i></a>, <a class="txt" href="matrlin.html#E8:51_1_2"><i><font color="Green" title="E44">A11</font></i></a>, <a class="txt" href="matrlin.html#E9:51_1_2"><i><font color="Green" title="E45">A12</font></i></a>, <a class="txt" href="matrlin.html#E12:51_1_2"><i><font color="Green" title="E47">A14</font></i></a>, <a class="ref" href="matrlin.html#T9" target="_self" title="MATRLIN:th.9">Th9</a></i>
<br/>.= 
<font color="Maroon" title="c6">F2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c10">k</font>
<b>by </b><i><a class="txt" href="matrlin.html#E1:51_1_2"><i><font color="Green" title="E39">A6</font></i></a>, <a class="txt" href="matrlin.html#E6:51_1_2"><i><font color="Green" title="E43">A10</font></i></a>, <a class="txt" href="matrlin.html#E10:51_1_2"><i><font color="Green" title="E46">A13</font></i></a>, <a class="txt" href="matrlin.html#E1:51_1_2_1"><i><font color="Green" title="E48">A15</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c6">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
<b>by </b><i><a class="txt" href="matrlin.html#E3:51_1_2_1"><i><font color="Green" title="E50">A17</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E14:51_1_2"/>
<font color="Maroon" title="c5">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F2</font>
 <b>by </b><i><a class="txt" href="matrlin.html#E1:51_1_2"><i><font color="Green" title="E39">A6</font></i></a>, <a class="txt" href="matrlin.html#E6:51_1_2"><i><font color="Green" title="E43">A10</font></i></a>, <a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a></i>;<br/>


</div>
