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<b>let </b><font color="Maroon" title="c1">K</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">K</font><br/> for <font color="Olive" title="b2">V1</font> being <a href="matrlin.html#V2" title="MATRLIN:attr.2">finite-dimensional</a> <a href="vectsp_1.html#NM7" title="VECTSP_1:NM.7">VectSp</a> of <font color="Maroon" title="c1">K</font><br/> for <font color="Olive" title="b3">b1</font> being   <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of <font color="Olive" title="b2">V1</font><br/> for <font color="Olive" title="b4">v1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b2">V1</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Olive" title="b4">v1</font></span>)</span> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Olive" title="b3">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">v1</font> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Olive" title="b3">b1</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">K</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">V1</font> being <a href="matrlin.html#V2" title="MATRLIN:attr.2">finite-dimensional</a> <a href="vectsp_1.html#NM7" title="VECTSP_1:NM.7">VectSp</a> of <font color="Maroon" title="c1">K</font><br/> for <font color="Olive" title="b2">b1</font> being   <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of <font color="Olive" title="b1">V1</font><br/> for <font color="Olive" title="b3">v1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b1">V1</font> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Olive" title="b3">v1</font></span>)</span> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Olive" title="b2">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">v1</font> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Olive" title="b2">b1</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c3">V1</font> be  <a href="matrlin.html#V2" title="MATRLIN:attr.2">finite-dimensional</a> <a href="vectsp_1.html#NM7" title="VECTSP_1:NM.7">VectSp</a> of <font color="Maroon" title="c1">K</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">b1</font> being   <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of <font color="Maroon" title="c3">V1</font><br/> for <font color="Olive" title="b2">v1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">V1</font> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Olive" title="b2">v1</font></span>)</span> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Olive" title="b1">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">v1</font> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Olive" title="b1">b1</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c4">b1</font> be    <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of <font color="Maroon" title="c3">V1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">v1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">V1</font> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Olive" title="b1">v1</font></span>)</span> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Maroon" title="c4">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">v1</font> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Maroon" title="c4">b1</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c5">v1</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">V1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c5">v1</font></span>)</span> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Maroon" title="c4">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Maroon" title="c4">b1</font></span>)</span></span><br/>









<b>set </b><font color="Maroon" title="c6">vb</font> = <font color="Maroon" title="c5">v1</font> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Maroon" title="c4">b1</font>;<br/>
<b>set </b><font color="Maroon" title="c7">avb</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c5">v1</font></span>)</span> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Maroon" title="c4">b1</font>;<br/>
<b>consider </b><font color="Maroon" title="c8">L1</font> being    <a href="vectsp_6.html#M1" title="VECTSP_6:mode.1">Linear_Combination</a> of <font color="Maroon" title="c3">V1</font><b> such that </b><br/><a NAME="E2:21"/><i><font color="Green" title="E11">A1</font></i>: 
( <font color="Maroon" title="c5">v1</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">L1</font> &amp;  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c8">L1</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="c4">b1</font> )
 <b>and </b><br/><a NAME="E3:21"/><i><font color="Green" title="E12">A2</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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Maroon" title="c4">b1</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Maroon" title="c4">b1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">L1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="matrlin.html#D9" title="MATRLIN:def.9">MATRLIN:def 9</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">L2</font> being    <a href="vectsp_6.html#M1" title="VECTSP_6:mode.1">Linear_Combination</a> of <font color="Maroon" title="c3">V1</font><b> such that </b><br/><a NAME="E5:21"/><i><font color="Green" title="E13">A3</font></i>: 
( <font color="Maroon" title="c2">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c5">v1</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="c9">L2</font> &amp;  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c9">L2</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="c4">b1</font> )
 <b>and </b><br/><a NAME="E6:21"/><i><font color="Green" title="E14">A4</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c5">v1</font></span>)</span> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Maroon" title="c4">b1</font></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c5">v1</font></span>)</span> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Maroon" title="c4">b1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">L2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="matrlin.html#D9" title="MATRLIN:def.9">MATRLIN:def 9</a></i>;<br/>
<a NAME="E7:21"/><i><font color="Green" title="E15">A5</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Maroon" title="c4">b1</font></span>)</span> <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="c4">b1</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c5">v1</font></span>)</span> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Maroon" title="c4">b1</font></span>)</span> <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="c4">b1</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Maroon" title="c4">b1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Maroon" title="c4">b1</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="matrlin.html#D9" title="MATRLIN:def.9">MATRLIN:def 9</a>, <a class="ref" href="matrixr1.html#T16" title="MATRIXR1:th.16">MATRIXR1:16</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c10">vb'</font> = <font color="Maroon" title="c5">v1</font> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Maroon" title="c4">b1</font>, <font color="Maroon" title="c11">avb</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c5">v1</font></span>)</span> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Maroon" title="c4">b1</font>, <font color="Maroon" title="c12">Avb</font> = <font color="Maroon" title="c2">a</font> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Maroon" title="c4">b1</font></span>)</span> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">b1</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">K</font> <b>by </b><i><a class="ref" href="finseq_2.html#T110" title="FINSEQ_2:th.110">FINSEQ_2:110</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">rb1</font> =  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">b1</font> as    <a href="vectsp_7.html#M1" title="VECTSP_7:mode.1">Basis</a> of <font color="Maroon" title="c3">V1</font> <b>by </b><i><a class="ref" href="matrlin.html#D4" title="MATRLIN:def.4">MATRLIN:def 4</a></i>;<br/>
<a NAME="E10:21"/><i><font color="Green" title="E16">A6</font></i>: 
<font color="Maroon" title="c13">rb1</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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c14">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c14">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">b1</font></span>)</span> implies <font color="Maroon" title="c11">avb</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">Avb</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font> )</span><br/><b>assume </b><a NAME="E1:21_1"/><i><font color="Green" title="E17">A7</font></i>: 
<font color="Maroon" title="c14">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">b1</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c11">avb</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">Avb</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font></span><br/><a NAME="E2:21_1"/><i><font color="Green" title="E18">A8</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">i</font> &amp; <font color="Maroon" title="c14">i</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="c4">b1</font> )
 <b>by </b><i><a class="txt" href="matrlin2.html#E1:21_1"><i><font color="Green" title="E17">A7</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E3:21_1"/>
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Maroon" title="c4">b1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">b1</font> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">avb</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">b1</font> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">b1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="matrlin2.html#E7:21"><i><font color="Green" title="E15">A5</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/><a NAME="E4:21_1"/><b>then </b><i><font color="Green" title="E19">A9</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Maroon" title="c4">b1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Maroon" title="c4">b1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c14">i</font> &amp; <font color="Maroon" title="c11">avb</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">avb</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c14">i</font> )
 <b>by </b><i><a class="txt" href="matrlin2.html#E1:21_1"><i><font color="Green" title="E17">A7</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><div><i><font color="Green" title="E20">A10</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:21_1_1_1"/>
( <font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font> or <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_1_1"><b>suppose </b></a><a NAME="E1:21_1_1_1_1"/>
<font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">L2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c14">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">vb'</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c14">i</font></span>)</span></span><br/><div class="add"><a NAME="E2:21_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c2">a</font> <a href="vectsp_6.html#K6" title="VECTSP_6:func.6">*</a> <font color="Maroon" title="c8">L1</font> <a href="vectsp_6.html#R1" title="VECTSP_6:pred.1">=</a> <font color="Maroon" title="c9">L2</font>
 <b>by </b><i><a class="txt" href="matrlin2.html#E2:21"><i><font color="Green" title="E11">A1</font></i></a>, <a class="txt" href="matrlin2.html#E5:21"><i><font color="Green" title="E13">A3</font></i></a>, <a class="txt" href="matrlin2.html#E10:21"><i><font color="Green" title="E16">A6</font></i></a>, <a class="ref" href="matrlin.html#T11" title="MATRLIN:th.11">MATRLIN:11</a></i>;<br/><b>hence </b><font color="Maroon" title="c9">L2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c14">i</font></span>)</span> = 
<font color="Maroon" title="c2">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">L1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c14">i</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_6.html#D12" title="VECTSP_6:def.12">VECTSP_6:def 12</a></i>
<br/>.= 
<font color="Maroon" title="c2">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">vb'</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c14">i</font></span>)</span>
<b>by </b><i><a class="txt" href="matrlin2.html#E7:21"><i><font color="Green" title="E15">A5</font></i></a>, <a class="txt" href="matrlin2.html#E2:21_1"><i><font color="Green" title="E18">A8</font></i></a>, <a class="txt" href="matrlin2.html#E3:21"><i><font color="Green" title="E12">A2</font></i></a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_1_2"><b>suppose </b></a><a NAME="E1:21_1_1_1_2"/><i><font color="Green" title="E21">A11</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">L2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c14">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">vb'</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c14">i</font></span>)</span></span><br/><div class="add"><a NAME="E2:21_1_1_1_2"/><b>then </b><i><font color="Green" title="E22">A12</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c5">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">V1</font>
 <b>by </b><i><a class="ref" href="vectsp_1.html#T59" title="VECTSP_1:th.59">VECTSP_1:59</a></i>;<br/><a NAME="E3:21_1_1_1_2"/><i><font color="Green" title="E23">A13</font></i>: 
<font color="Maroon" title="c9">L2</font> is    <a href="vectsp_6.html#M2" title="VECTSP_6:mode.2">Linear_Combination</a> of  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c9">L2</font>
 <b>by </b><i><a class="ref" href="vectsp_6.html#T28" title="VECTSP_6:th.28">VECTSP_6:28</a></i>;<br/><a NAME="E4:21_1_1_1_2"/>
 <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c9">L2</font> is <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a>
 <b>by </b><i><a class="txt" href="matrlin2.html#E5:21"><i><font color="Green" title="E13">A3</font></i></a>, <a class="ref" href="vectsp_7.html#T2" title="VECTSP_7:th.2">VECTSP_7:2</a>, <a class="txt" href="matrlin2.html#E10:21"><i><font color="Green" title="E16">A6</font></i></a></i>;<br/><a NAME="E5:21_1_1_1_2"/><b>then </b>
not <font color="Maroon" title="c4">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c14">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c9">L2</font>
 <b>by </b><i><a class="txt" href="matrlin2.html#E5:21"><i><font color="Green" title="E13">A3</font></i></a>, <a class="txt" href="matrlin2.html#E2:21_1_1_1_2"><i><font color="Green" title="E22">A12</font></i></a>, <a class="txt" href="matrlin2.html#E3:21_1_1_1_2"><i><font color="Green" title="E23">A13</font></i></a>, <a class="ref" href="vectsp_7.html#D1" title="VECTSP_7:def.1">VECTSP_7:def 1</a></i>;<br/><b>hence </b><font color="Maroon" title="c9">L2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c14">i</font></span>)</span> = 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font>

<br/>.= 
<font color="Maroon" title="c2">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">vb'</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c14">i</font></span>)</span>
<b>by </b><i><a class="txt" href="matrlin2.html#E1:21_1_1_1_2"><i><font color="Green" title="E21">A11</font></i></a>, <a class="ref" href="vectsp_1.html#T39" title="VECTSP_1:th.39">VECTSP_1:39</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>thus </b><font color="Maroon" title="c11">avb</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font> = 
<font color="Maroon" title="c9">L2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c14">i</font></span>)</span>
<b>by </b><i><a class="txt" href="matrlin2.html#E4:21_1"><i><font color="Green" title="E19">A9</font></i></a>, <a class="txt" href="matrlin2.html#E7:21"><i><font color="Green" title="E15">A5</font></i></a>, <a class="txt" href="matrlin2.html#E6:21"><i><font color="Green" title="E14">A4</font></i></a>, <a class="txt" href="matrlin2.html#E2:21_1"><i><font color="Green" title="E18">A8</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c12">Avb</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font>
<b>by </b><i><a class="txt" href="matrlin2.html#E5:21_1"><i><font color="Green" title="E20">A10</font></i></a>, <a class="txt" href="matrlin2.html#E1:21_1"><i><font color="Green" title="E17">A7</font></i></a>, <a class="txt" href="matrlin2.html#E4:21_1"><i><font color="Green" title="E19">A9</font></i></a>, <a class="ref" href="fvsum_1.html#T63" title="FVSUM_1:th.63">FVSUM_1:63</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E12:21"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c5">v1</font></span>)</span> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Maroon" title="c4">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Maroon" title="c4">b1</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_2.html#T139" title="FINSEQ_2:th.139">FINSEQ_2:139</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
