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

<b>let </b><font color="Maroon">K</font> be   <a href="vectsp_1.html#NM2">Field</a>;<br/><b>let </b><font color="Maroon">V1</font> be  <a href="matrlin.html#V2" target="_self">finite-dimensional</a> <a href="vectsp_1.html#NM6">VectSp</a> of <font color="Maroon">K</font>;<br/><b>let </b><font color="Maroon">b1</font> be    <a href="matrlin.html#M1" target="_self">OrdBasis</a> of <font color="Maroon">V1</font>;<br/><b>let </b><font color="Maroon">d</font> be   <a href="struct_0.html#NM7">FinSequence</a> of <font color="Maroon">K</font>;<br/>







<b>assume </b><a NAME="E1:55"/><i><font color="Green">E43</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">d</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">b1</font>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V1</font>,  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">K</font>] <b>means </b>( ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">b1</font> implies for <font color="Olive">k</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">b1</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">k</font> ) &amp; ( not <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">b1</font> implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font> ) );<br/>
<a NAME="E2:55"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">v</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V1</font>  ex <font color="Olive">u</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">K</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">v</font>,<font color="Olive">u</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">v</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V1</font>;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:55_1_1"/>
( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">b1</font> or  not <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">b1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_1_1"><b>suppose </b></a><a NAME="E1:55_1_1_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">b1</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:55_1_1_1"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">b1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> )
 <b>by </b><i><a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/><b>take </b><font color="Maroon">u</font> = <font color="Maroon">d</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:55_1_1_1_1"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">b1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> )
 ;<br/><a NAME="E2:55_1_1_1_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">b1</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i/>;<br/><font color="Maroon">b1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
<font color="Maroon">b1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font>
<b>by </b><i><a class="ref" href="matrlin.html#T4" target="_self">Th4</a>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon">b1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
<b>by </b><i><a class="ref" href="matrlin.html#T4" target="_self">Th4</a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/><b>hence </b><a NAME="E4:55_1_1_1_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="matrlin.html#T4" target="_self">Th4</a>, , , <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:55_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">v</font>,<font color="Maroon">u</font>]
 <b>by </b><i><a class="ref" href="matrlin.html#T3" target="_self">Th3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_1_2"><b>suppose </b></a><a NAME="E1:55_1_1_2"/><i><font color="Green">E57</font></i>: 
not <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">b1</font>
 ;<br/><div class="add"><b>take </b>
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font>
;<br/><b>thus </b><a NAME="E2:55_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">v</font>, <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font>]
 <b>by </b><i><a class="ref" href="matrlin.html#T3" target="_self">Th3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>consider </b><font color="Maroon">KL</font> being   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">V1</font>,<font color="Maroon">K</font><b> such that </b><br/><a NAME="E4:55"/><i><font color="Green">E58</font></i>: 
for <font color="Olive">v</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V1</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">v</font>,<font color="Maroon">KL</font> <a href="funct_2.html#K3">.</a> <font color="Olive">v</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>();<br/></i>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_2"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">f</font> = <font color="Maroon">KL</font>;<br/><b>thus </b><a NAME="E1:55_2"/>
( <font color="Maroon">KL</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">f</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V1</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">K</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/></div><b>end;</b></div>
<a NAME="E6:55"/><b>then </b><i><font color="Green">E65</font></i>: 
<font color="Maroon">KL</font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">K</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">T</font> =  <a href="relat_1.html#K2">rng</a> <font color="Maroon">b1</font> as  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V1</font> <b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_3"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">T</font> = <font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">v</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V1</font>;<br/><b>assume </b><a NAME="E1:55_3"/>
not <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font>
 ;<br/><b>hence </b><a NAME="E2:55_3"/>
<font color="Maroon">KL</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font>
 <b>by </b><i><a class="ref" href="matrlin.html#T3" target="_self">Th3</a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">KL1</font> = <font color="Maroon">KL</font> as    <a href="vectsp_6.html#M1">Linear_Combination</a> of <font color="Maroon">V1</font> <b>by </b><i><a class="ref" href="matrlin.html#T4" target="_self">Th4</a>, <a class="ref" href="vectsp_6.html#D4">VECTSP_6:def 4</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_4"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">KL1</font> = <font color="Maroon">KL1</font>;<br/><b>thus </b><a NAME="E1:55_4"/><i><font color="Green">E66</font></i>: 
for <font color="Olive">k</font> being  <a href="nat_1.html#NM1">Nat</a>  st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">d</font> holds <br/><font color="Maroon">d</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">KL1</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">k</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:55_4_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">d</font> )
 ;<br/>

<a NAME="E2:55_4_1"/><b>then </b><i><font color="Green">E67</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">b1</font>
 
<b>by </b><i><a class="txt" href="matrlin.html#E2:55"><i><font color="Green">E44</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E3:55_4_1"/><b>then </b>
<font color="Maroon">b1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font>
 
<b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E4:55_4_1"/><b>then </b>
<font color="Maroon">b1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">b1</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E5:55_4_1"/>
<font color="Maroon">d</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">KL1</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="matrlin.html#T3" target="_self">Th3</a>, </i>;<br/>


</div><b>end;</b></div><a NAME="E2:55_4"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">KL1</font> holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">b1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_4_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:55_4_2"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">KL1</font>
 ;<br/>

<b>assume </b><a NAME="E2:55_4_2"/><i><font color="Green">E69</font></i>: 
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">b1</font>
 ;<br/>

<b>consider </b><font color="Maroon">v</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V1</font><b> such that </b><br/><a NAME="E4:55_4_2"/><i><font color="Green">E111</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> &amp; <font color="Maroon">KL1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font> )
 <b>by </b><i>, <a class="ref" href="vectsp_6.html#T19">VECTSP_6:19</a></i>;<br/>
<b>thus </b><a NAME="E5:55_4_2"/>
contradiction
 <b>by </b><i><a class="ref" href="matrlin.html#T3" target="_self">Th3</a>, , </i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:55_4"/><i><font color="Green">E112</font></i>: 
 <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">KL1</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">b1</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E4:55_4"/><i><font color="Green">E113</font></i>: 
<font color="Maroon">b1</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i/>;<br/><a NAME="E5:55_4"/><i><font color="Green">E134</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">d</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">b1</font>
 <b>by </b><i><a class="txt" href="matrlin.html#E2:55"><i><font color="Green">E44</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><a NAME="E6:55_4"/><b>then </b><i><font color="Green">E135</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="matrlin.html#K7" target="_self">lmlt</a> <font color="Maroon">d</font>,<font color="Maroon">b1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">b1</font>
 <b>by </b><i/>;<br/><b>then </b><i><font color="Green">E136</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrlin.html#K7" target="_self">lmlt</a> <font color="Maroon">d</font>,<font color="Maroon">b1</font></span>)</span> = 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">b1</font>
<b>by </b><i><a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">KL1</font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">b1</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_6.html#D8">VECTSP_6:def 8</a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_4_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:55_4_4"/><i><font color="Green">E137</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="matrlin.html#K7" target="_self">lmlt</a> <font color="Maroon">d</font>,<font color="Maroon">b1</font></span>)</span>
 ;<br/><a NAME="E2:55_4_4"/><b>then </b><i><font color="Green">E154</font></i>: 
<font color="Maroon">d</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E3:55_4_4"/><i><font color="Green">E155</font></i>: 
<font color="Maroon">b1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E4:55_4_4"/><i><font color="Green">E156</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">KL1</font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">b1</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><a NAME="E5:55_4_4"/><i><font color="Green">E158</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">d</font> )
 <b>by </b><i><a class="txt" href="matrlin.html#E2:55"><i><font color="Green">E44</font></i></a>, , , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="matrlin.html#K7" target="_self">lmlt</a> <font color="Maroon">d</font>,<font color="Maroon">b1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> = 
the <a href="vectsp_1.html#U2">lmult</a> of <font color="Maroon">V1</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i>, <a class="ref" href="funcop_1.html#T28">FUNCOP_1:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="vectsp_1.html#D24">VECTSP_1:def 24</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">KL1</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="vectsp_1.html#K6">*</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i>, <a class="txt" href="matrlin.html#E4:55"><i><font color="Green">E58</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">KL1</font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">b1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
<b>by </b><i>, <a class="ref" href="vectsp_6.html#D8">VECTSP_6:def 8</a></i>
;<br/></div><b>end;</b></div><b>hence </b> <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K7" target="_self">lmlt</a> <font color="Maroon">d</font>,<font color="Maroon">b1</font></span>)</span> = 
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">KL1</font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">b1</font></span>)</span>
<b>by </b><i>, <a class="ref" href="newton.html#T3">NEWTON:3</a></i>
<br/>.= 
 <a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">KL1</font>
<b>by </b><i>, , </i>
;<br/><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E11:55"/>
<font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p2">(<span class="default"><a href="matrlin.html#K7" target="_self">lmlt</a> <font color="Maroon">d</font>,<font color="Maroon">b1</font></span>)</span></span>)</span> <a href="matrlin.html#K11" target="_self">|--</a> <font color="Maroon">b1</font>
 <b>by </b><i><a class="txt" href="matrlin.html#E2:55"><i><font color="Green">E44</font></i></a>, </i>;<br/>


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