<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">K</font> be   <a href="vectsp_1.html#NM2" title="VECTSP_1:NM.2">Field</a>;<br/><b>let </b><font color="Maroon" title="c2">V1</font> be  <a href="matrlin.html#V2" title="MATRLIN:attr.2">finite-dimensional</a> <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">VectSp</a> of <font color="Maroon" title="c1">K</font>;<br/><b>let </b><font color="Maroon" title="c3">b1</font> be    <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of <font color="Maroon" title="c2">V1</font>;<br/><b>let </b><font color="Maroon" title="c4">v</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">V1</font>;<br/>







<b>consider </b><font color="Maroon" title="c5">KL</font> being    <a href="vectsp_6.html#M1" title="VECTSP_6:mode.1">Linear_Combination</a> of <font color="Maroon" title="c2">V1</font><b> such that </b><br/><a NAME="E2:54"/><i><font color="Green" title="E43">A1</font></i>: 
( <font color="Maroon" title="c4">v</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="c5">KL</font> &amp;  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c5">KL</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> &amp; ( 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="c4">v</font> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Maroon" title="c3">b1</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Maroon" title="c3">b1</font></span>)</span> <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="c5">KL</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> ) )
 <b>by </b><i><a class="ref" href="matrlin.html#D9" target="_self" title="MATRLIN:def.9">Def9</a></i>;<br/>
<a NAME="E3:54"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Maroon" title="c3">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="c3">b1</font>
 
<b>by </b><i><a class="ref" href="matrlin.html#D9" target="_self" title="MATRLIN:def.9">Def9</a></i>;<br/>
<a NAME="E4:54"/><b>then </b><i><font color="Green" title="E44">A2</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Maroon" title="c3">b1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">b1</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E5:54"/><b>then </b><i><font color="Green" title="E45">A3</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="matrlin.html#K7" title="MATRLIN:func.7">lmlt</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Maroon" title="c3">b1</font></span>)</span>,<font color="Maroon" title="c3">b1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="matrlin.html#T16" target="_self" title="MATRLIN:th.16">Th16</a></i>;<br/>
<i><font color="Green" title="E46">A4</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">KL</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <font color="Maroon" title="c3">b1</font></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">b1</font>
<b>by </b><i><a class="ref" href="vectsp_6.html#D8" title="VECTSP_6:def.8">VECTSP_6:def 8</a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrlin.html#K7" title="MATRLIN:func.7">lmlt</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Maroon" title="c3">b1</font></span>)</span>,<font color="Maroon" title="c3">b1</font></span>)</span>
<b>by </b><i><a class="txt" href="matrlin.html#E5:54"><i><font color="Green" title="E45">A3</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>
;<br/>
<div><i><font color="Green" title="E47">A5</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="54_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">t</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="E1:54_2"/><i><font color="Green" title="E48">A6</font></i>: 
<font color="Maroon" title="c6">t</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> <span class="p1">(<span class="default"><a href="matrlin.html#K7" title="MATRLIN:func.7">lmlt</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Maroon" title="c3">b1</font></span>)</span>,<font color="Maroon" title="c3">b1</font></span>)</span>
 ;<br/><a NAME="E2:54_2"/><b>then </b><i><font color="Green" title="E49">A7</font></i>: 
<font color="Maroon" title="c6">t</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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">KL</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <font color="Maroon" title="c3">b1</font></span>)</span>
 <b>by </b><i><a class="txt" href="matrlin.html#E6:54"><i><font color="Green" title="E46">A4</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/><a NAME="E3:54_2"/>
<font color="Maroon" title="c6">t</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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Maroon" title="c3">b1</font></span>)</span>
 <b>by </b><i><a class="txt" href="matrlin.html#E4:54"><i><font color="Green" title="E44">A2</font></i></a>, <a class="txt" href="matrlin.html#E1:54_2"><i><font color="Green" title="E48">A6</font></i></a>, <a class="ref" href="matrlin.html#T16" target="_self" title="MATRLIN:th.16">Th16</a></i>;<br/><a NAME="E4:54_2"/><b>then </b><i><font color="Green" title="E50">A8</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">t</font> &amp; <font color="Maroon" title="c6">t</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="c4">v</font> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Maroon" title="c3">b1</font></span>)</span> )
 <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="E5:54_2"/><i><font color="Green" title="E51">A9</font></i>: 
<font color="Maroon" title="c3">b1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c6">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">t</font>
 <b>by </b><i><a class="txt" href="matrlin.html#E5:54"><i><font color="Green" title="E45">A3</font></i></a>, <a class="txt" href="matrlin.html#E1:54_2"><i><font color="Green" title="E48">A6</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E6:54_2"/><i><font color="Green" title="E52">A10</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Maroon" title="c3">b1</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c6">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Maroon" title="c3">b1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">t</font>
 <b>by </b><i><a class="txt" href="matrlin.html#E4:54_2"><i><font color="Green" title="E50">A8</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c5">KL</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <font color="Maroon" title="c3">b1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">t</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">KL</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<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="c6">t</font></span>)</span></span>)</span> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</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="c6">t</font></span>)</span>
<b>by </b><i><a class="txt" href="matrlin.html#E2:54_2"><i><font color="Green" title="E49">A7</font></i></a>, <a class="ref" href="vectsp_6.html#D8" title="VECTSP_6:def.8">VECTSP_6:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Maroon" title="c3">b1</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c6">t</font></span>)</span> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</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="c6">t</font></span>)</span>
<b>by </b><i><a class="txt" href="matrlin.html#E2:54"><i><font color="Green" title="E43">A1</font></i></a>, <a class="txt" href="matrlin.html#E4:54_2"><i><font color="Green" title="E50">A8</font></i></a></i>
<br/>.= 
the <a href="vectsp_1.html#U2" title="VECTSP_1:sel.2">lmult</a> of <font color="Maroon" title="c2">V1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Maroon" title="c3">b1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">t</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">t</font></span>)</span>
<b>by </b><i><a class="txt" href="matrlin.html#E5:54_2"><i><font color="Green" title="E51">A9</font></i></a>, <a class="txt" href="matrlin.html#E6:54_2"><i><font color="Green" title="E52">A10</font></i></a>, <a class="ref" href="vectsp_1.html#D24" title="VECTSP_1:def.24">VECTSP_1:def 24</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrlin.html#K7" title="MATRLIN:func.7">lmlt</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Maroon" title="c3">b1</font></span>)</span>,<font color="Maroon" title="c3">b1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">t</font>
<b>by </b><i><a class="txt" href="matrlin.html#E1:54_2"><i><font color="Green" title="E48">A6</font></i></a>, <a class="ref" href="funcop_1.html#T28" title="FUNCOP_1:th.28">FUNCOP_1:28</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E8:54"/>
<font color="Maroon" title="c3">b1</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="matrlin.html#D4" target="_self" title="MATRLIN:def.4">Def4</a></i>;<br/>
<b>hence </b><font color="Maroon" title="c4">v</font> = 
 <a href="rlvect_1.html#K9" title="RLVECT_1:func.9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">KL</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <font color="Maroon" title="c3">b1</font></span>)</span>
<b>by </b><i><a class="txt" href="matrlin.html#E2:54"><i><font color="Green" title="E43">A1</font></i></a>, <a class="ref" href="matrlin.html#T24" target="_self" title="MATRLIN:th.24">Th24</a></i>
<br/>.= 
 <a href="rlvect_1.html#K9" title="RLVECT_1:func.9">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K7" title="MATRLIN:func.7">lmlt</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Maroon" title="c3">b1</font></span>)</span>,<font color="Maroon" title="c3">b1</font></span>)</span>
<b>by </b><i><a class="txt" href="matrlin.html#E6:54"><i><font color="Green" title="E46">A4</font></i></a>, <a class="txt" href="matrlin.html#E7:54"><i><font color="Green" title="E47">A5</font></i></a>, <a class="ref" href="newton.html#T3" title="NEWTON:th.3">NEWTON:3</a></i>
;<br/><br/>


</div>
