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

<b>let </b><font color="Maroon">R</font> be   <a href="vectsp_2.html#NM2">domRing</a>;<br/><b>let </b><font color="Maroon">V</font> be   <a href="vectsp_2.html#NM7">RightMod</a> of <font color="Maroon">R</font>;<br/><b>let </b><font color="Maroon">L</font> be    <a href="rmod_4.html#M1" target="_self">Linear_Combination</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="vectsp_1.html#NM3">Scalar</a> of <font color="Maroon">R</font>;<br/>







<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:70_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font> or <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_1"><b>suppose </b></a><a NAME="E1:70_1_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon">F</font> being   <a href="struct_0.html#NM7">FinSequence</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E3:70_1_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">F</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>and </b><br/><a NAME="E4:70_1_1"/><i><font color="Green">E50</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="rmod_4.html#K2" target="_self">Carrier</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="rmod_4.html#K7" target="_self">*</a> <font color="Maroon">a</font></span>)</span>
 <b>and </b><br/><a NAME="E5:70_1_1"/><i><font color="Green">E52</font></i>: 
 <a href="rmod_4.html#K5" target="_self">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="rmod_4.html#K7" target="_self">*</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="rmod_4.html#K7" target="_self">*</a> <font color="Maroon">a</font></span>)</span> <a href="rmod_4.html#K4" target="_self">(#)</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">G</font> being   <a href="struct_0.html#NM7">FinSequence</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E7:70_1_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">G</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>and </b><br/><a NAME="E8:70_1_1"/><i><font color="Green">E54</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="rmod_4.html#K2" target="_self">Carrier</a> <font color="Maroon">L</font>
 <b>and </b><br/><a NAME="E9:70_1_1"/><i><font color="Green">E55</font></i>: 
 <a href="rmod_4.html#K5" target="_self">Sum</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="rmod_4.html#K4" target="_self">(#)</a> <font color="Maroon">G</font></span>)</span>
 <b>by </b><i/>;<br/><b>set </b><font color="Maroon">g</font> = <font color="Maroon">L</font> <a href="rmod_4.html#K4" target="_self">(#)</a> <font color="Maroon">G</font>;<br/><b>set </b><font color="Maroon">f</font> = <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="rmod_4.html#K7" target="_self">*</a> <font color="Maroon">a</font></span>)</span> <a href="rmod_4.html#K4" target="_self">(#)</a> <font color="Maroon">F</font>;<br/><b>set </b><font color="Maroon">l</font> = <font color="Maroon">L</font> <a href="rmod_4.html#K7" target="_self">*</a> <font color="Maroon">a</font>;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="nat_1.html#NM1">Nat</a>) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">F</font> <a href="finseq_4.html#K1">&lt;-</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/><b>consider </b><font color="Maroon">P</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E11:70_1_1"/><i><font color="Green">E56</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font>
 <b>and </b><br/><a NAME="E12:70_1_1"/><i><font color="Green">E57</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> holds <br/><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">k</font>)
 <b>from </b><i><a class="ref" href="finseq_1.html#S2">FINSEQ_1:sch 2</a>();<br/></i><a NAME="E13:70_1_1"/><i><font color="Green">E58</font></i>: 
 <a href="rmod_4.html#K2" target="_self">Carrier</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="rmod_4.html#K7" target="_self">*</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rmod_4.html#K2" target="_self">Carrier</a> <font color="Maroon">L</font>
 <b>by </b><i>, </i>;<br/><a NAME="E14:70_1_1"/><b>then </b><i><font color="Green">E59</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font>
 <b>by </b><i>, , <a class="ref" href="rmod_4.html#T1" target="_self">Th1</a>, , <a class="ref" href="finseq_1.html#T65">FINSEQ_1:65</a></i>;<br/><a NAME="E15:70_1_1"/><i><font color="Green">E60</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">P</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:70_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">P</font>
 ;<br/>

<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:70_1_1_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">P</font>
 <b>and </b><br/><a NAME="E4:70_1_1_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="rmod_4.html#E7:70_1_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="finseq_3.html#T25">FINSEQ_3:25</a></i>;<br/>
<a NAME="E6:70_1_1_1"/><i><font color="Green">E63</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E7:70_1_1_1"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="txt" href="rmod_4.html#E1:70_1_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="rmod_4.html#E7:70_1_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E8:70_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">G</font>
 
<b>by </b><i><a class="txt" href="rmod_4.html#E1:70_1_1"><i><font color="Green">E48</font></i></a>, , <a class="txt" href="rmod_4.html#E7:70_1_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E9:70_1_1_1"/><b>then </b>
<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font>
 
<b>by </b><i>, , <a class="txt" href="rmod_4.html#E4:70_1_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E10:70_1_1_1"/><b>then </b>
( <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="finseq_4.html#K1">&lt;-</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font></span>)</span> &amp; <font color="Maroon">F</font> <a href="finseq_4.html#R2">just_once_values</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> )
 
<b>by </b><i>, <a class="txt" href="rmod_4.html#E3:70_1_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="rmod_4.html#E9:70_1_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="rmod_4.html#E11:70_1_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="finseq_4.html#T10">FINSEQ_4:10</a></i>;<br/>
<b>hence </b><a NAME="E11:70_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="txt" href="rmod_4.html#E8:70_1_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="finseq_4.html#D3">FINSEQ_4:def 3</a></i>;<br/>


</div><b>end;</b></div><div><i><font color="Green">E99</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>thus </b><a NAME="E1:70_1_1_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">G</font> implies ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">P</font> &amp; <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:70_1_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">G</font>
 ;<br/>

<a NAME="E2:70_1_1_2_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span>
 
<b>by </b><i><a class="txt" href="rmod_4.html#E1:70_1_1"><i><font color="Green">E48</font></i></a>, , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E3:70_1_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>

<a NAME="E4:70_1_1_2_1"/><b>then </b>
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">P</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> &amp; <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">P</font> )
 
<b>by </b><i><a class="txt" href="rmod_4.html#E5:70_1_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E5:70_1_1_2_1"/>
<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font>
 ;<br/>


</div><b>end;</b></div><b>assume </b><a NAME="E2:70_1_1_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">P</font> &amp; <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> )
 ;<br/><a NAME="E3:70_1_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>hence </b><a NAME="E4:70_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="txt" href="rmod_4.html#E1:70_1_1"><i><font color="Green">E48</font></i></a>, , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_1_3"><b>now </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="E1:70_1_1_3"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">G</font>
 ;<br/><b>then reconsider </b><font color="Maroon">n</font> = <font color="Maroon">x</font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="ref" href="finseq_3.html#T25">FINSEQ_3:25</a></i>;<br/><a NAME="E3:70_1_1_3"/>
<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font>
 <b>by </b><i>, , <a class="txt" href="rmod_4.html#E4:70_1_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="rmod_4.html#E8:70_1_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E4:70_1_1_3"/><b>then </b><i><font color="Green">E101</font></i>: 
<font color="Maroon">F</font> <a href="finseq_4.html#R2">just_once_values</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="finseq_4.html#T10">FINSEQ_4:10</a></i>;<br/><a NAME="E5:70_1_1_3"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="rmod_4.html#E8:70_1_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>then </b><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> = 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_4.html#K1">&lt;-</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="rmod_4.html#E3:70_1_1"><i><font color="Green">E49</font></i></a></i>
<br/>.= 
<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
<b>by </b><i><a class="txt" href="rmod_4.html#E9:70_1_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="finseq_4.html#D3">FINSEQ_4:def 3</a></i>
;<br/><b>hence </b><a NAME="E7:70_1_1_3"/>
<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>
 ;<br/></div><b>end;</b></div><a NAME="E18:70_1_1"/><b>then </b><i><font color="Green">E102</font></i>: 
<font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="txt" href="rmod_4.html#E7:70_1_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="funct_1.html#T20">FUNCT_1:20</a></i>;<br/><a NAME="E19:70_1_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">P</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:70_1_1_4"/><i><font color="Green">E105</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font>
 ;<br/>

<b>set </b><font color="Maroon">f</font> = <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">G</font>;<br/>
<a NAME="E2:70_1_1_4"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">G</font>
 
<b>by </b><i>, , , <a class="txt" href="rmod_4.html#E4:70_1_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="funct_1.html#T55">FUNCT_1:55</a></i>;<br/>
<b>then </b><i><font color="Green">E106</font></i>:  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">G</font></span>)</span> = 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K2">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T47">RELAT_1:47</a></i>
<br/>.= 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T55">FUNCT_1:55</a></i>
;<br/>
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">G</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">P</font>
<b>by </b><i><a class="txt" href="rmod_4.html#E8:70_1_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K6">id</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">P</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T61">FUNCT_1:61</a></i>
<br/>.= 
<font color="Maroon">P</font>
<b>by </b><i><a class="txt" href="rmod_4.html#E5:70_1_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="relat_1.html#T79">RELAT_1:79</a></i>
;<br/>
<b>hence </b><a NAME="E5:70_1_1_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="txt" href="rmod_4.html#E9:70_1_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="rmod_4.html#E11:70_1_1"><i><font color="Green">E56</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E20:70_1_1"/><b>then </b><i><font color="Green">E107</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="txt" href="rmod_4.html#E5:70_1_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/><a NAME="E21:70_1_1"/><i><font color="Green">E108</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="txt" href="rmod_4.html#E1:70_1_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><a NAME="E22:70_1_1"/><b>then </b><i><font color="Green">E109</font></i>: 
<font color="Maroon">P</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="txt" href="rmod_4.html#E9:70_1_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="finseq_4.html#T75">FINSEQ_4:75</a></i>;<br/><a NAME="E23:70_1_1"/><i><font color="Green">E110</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="rmod_4.html#K7" target="_self">*</a> <font color="Maroon">a</font></span>)</span> <a href="rmod_4.html#K4" target="_self">(#)</a> <font color="Maroon">F</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font>
 <b>by </b><i/>;<br/><a NAME="E24:70_1_1"/><b>then </b><i><font color="Green">E113</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="rmod_4.html#K7" target="_self">*</a> <font color="Maroon">a</font></span>)</span> <a href="rmod_4.html#K4" target="_self">(#)</a> <font color="Maroon">F</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><a NAME="E25:70_1_1"/>
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  implies  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/><b>then reconsider </b><font color="Maroon">P</font> = <font color="Maroon">P</font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font>, <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> <b>by </b><i><a class="txt" href="rmod_4.html#E5:70_1_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="rmod_4.html#E11:70_1_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/><b>reconsider </b><font color="Maroon">P</font> = <font color="Maroon">P</font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="rmod_4.html#K7" target="_self">*</a> <font color="Maroon">a</font></span>)</span> <a href="rmod_4.html#K4" target="_self">(#)</a> <font color="Maroon">F</font></span>)</span>, <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="rmod_4.html#K7" target="_self">*</a> <font color="Maroon">a</font></span>)</span> <a href="rmod_4.html#K4" target="_self">(#)</a> <font color="Maroon">F</font></span>)</span> <b>by </b><i><a class="txt" href="rmod_4.html#E14:70_1_1"><i><font color="Green">E59</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">Fp</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="rmod_4.html#K7" target="_self">*</a> <font color="Maroon">a</font></span>)</span> <a href="rmod_4.html#K4" target="_self">(#)</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">P</font> as   <a href="struct_0.html#NM7">FinSequence</a> of <font color="Maroon">V</font> <b>by </b><i><a class="ref" href="finseq_2.html#T51">FINSEQ_2:51</a></i>;<br/><b>reconsider </b><font color="Maroon">P</font> = <font color="Maroon">P</font> as   <a href="funct_2.html#NM2">Permutation</a> of  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="rmod_4.html#K7" target="_self">*</a> <font color="Maroon">a</font></span>)</span> <a href="rmod_4.html#K4" target="_self">(#)</a> <font color="Maroon">F</font></span>)</span> <b>by </b><i><a class="txt" href="rmod_4.html#E9:70_1_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="rmod_4.html#E12:70_1_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="rmod_4.html#E14:70_1_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="funct_2.html#T83">FUNCT_2:83</a></i>;<br/><a NAME="E30:70_1_1"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">Fp</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="rmod_4.html#K7" target="_self">*</a> <font color="Maroon">a</font></span>)</span> <a href="rmod_4.html#K4" target="_self">(#)</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">P</font>
 ;<br/><a NAME="E31:70_1_1"/><b>then </b><i><font color="Green">E115</font></i>: 
 <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">Fp</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="rmod_4.html#K7" target="_self">*</a> <font color="Maroon">a</font></span>)</span> <a href="rmod_4.html#K4" target="_self">(#)</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_2.html#T9">RLVECT_2:9</a></i>;<br/><a NAME="E32:70_1_1"/><i><font color="Green">E116</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">Fp</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="rmod_4.html#K7" target="_self">*</a> <font color="Maroon">a</font></span>)</span> <a href="rmod_4.html#K4" target="_self">(#)</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i><a class="txt" href="rmod_4.html#E15:70_1_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="finseq_2.html#T48">FINSEQ_2:48</a></i>;<br/><a NAME="E33:70_1_1"/><b>then </b><i><font color="Green">E117</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">Fp</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="rmod_4.html#K4" target="_self">(#)</a> <font color="Maroon">G</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="rmod_4.html#E13:70_1_1"><i><font color="Green">E58</font></i></a>, </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_1_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">v</font> be   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">V</font>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:70_1_1_5"/><i><font color="Green">E118</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">Fp</font>
 <b>and </b><br/><a NAME="E2:70_1_1_5"/><i><font color="Green">E119</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="rmod_4.html#K4" target="_self">(#)</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 ;<br/><a NAME="E3:70_1_1_5"/><i><font color="Green">E120</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="rmod_4.html#K4" target="_self">(#)</a> <font color="Maroon">G</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="txt" href="rmod_4.html#E3"><i><font color="Green">Lemma64</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E4:70_1_1_5"/><b>then </b><i><font color="Green">E121</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">G</font>
 <b>by </b><i>, <a class="txt" href="rmod_4.html#E13:70_1_1"><i><font color="Green">E58</font></i></a>, , , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E5:70_1_1_5"/><b>then </b>
<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E6:70_1_1_5"/><b>then </b>
<font color="Maroon">F</font> <a href="finseq_4.html#R2">just_once_values</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, , , <a class="txt" href="rmod_4.html#E4:70_1_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="finseq_4.html#T10">FINSEQ_4:10</a></i>;<br/><a NAME="E7:70_1_1_5"/><b>then </b><i><font color="Green">E122</font></i>: 
<font color="Maroon">F</font> <a href="finseq_4.html#K1">&lt;-</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="finseq_4.html#D3">FINSEQ_4:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon">i</font> = <font color="Maroon">F</font> <a href="finseq_4.html#K1">&lt;-</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="ref" href="finseq_3.html#T25">FINSEQ_3:25</a></i>;<br/><a NAME="E9:70_1_1_5"/><i><font color="Green">E123</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">L</font> <a href="rmod_4.html#K4" target="_self">(#)</a> <font color="Maroon">G</font></span>)</span>
 <b>by </b><i><a class="ref" href="rmod_4.html#T5" target="_self">Th5</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E10:70_1_1_5"/><i><font color="Green">E124</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">P</font>
 <b>by </b><i><a class="txt" href="rmod_4.html#E1:70_1_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="rmod_4.html#E13:70_1_1"><i><font color="Green">E58</font></i></a>, , , <a class="ref" href="rmod_4.html#T5" target="_self">Th5</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><i><font color="Green">E125</font></i>: <font color="Maroon">G</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> = 
<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
<b>by </b><i><a class="ref" href="rmod_4.html#T9" target="_self">Th9</a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i><a class="txt" href="rmod_4.html#E8:70_1_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="rmod_4.html#T12" target="_self">Th12</a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i><a class="txt" href="rmod_4.html#E3:70_1_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="rmod_4.html#E13:70_1_1"><i><font color="Green">E58</font></i></a>, , , <a class="ref" href="rmod_4.html#T5" target="_self">Th5</a></i>
<br/>.= 
<font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>
<b>by </b><i><a class="ref" href="rmod_4.html#T10" target="_self">Th10</a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/><a NAME="E12:70_1_1_5"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="rmod_4.html#K7" target="_self">*</a> <font color="Maroon">a</font></span>)</span> <a href="rmod_4.html#K4" target="_self">(#)</a> <font color="Maroon">F</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="rmod_4.html#E13:70_1_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="rmod_4.html#T10" target="_self">Th10</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E13:70_1_1_5"/><b>then </b><i><font color="Green">E126</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="rmod_4.html#K7" target="_self">*</a> <font color="Maroon">a</font></span>)</span> <a href="rmod_4.html#K4" target="_self">(#)</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>thus </b><font color="Maroon">Fp</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="rmod_4.html#K7" target="_self">*</a> <font color="Maroon">a</font></span>)</span> <a href="rmod_4.html#K4" target="_self">(#)</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i><a class="ref" href="rmod_4.html#T12" target="_self">Th12</a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="rmod_4.html#K7" target="_self">*</a> <font color="Maroon">a</font></span>)</span> <a href="rmod_4.html#K4" target="_self">(#)</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_4.html#K1">&lt;-</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="rmod_4.html#E3:70_1_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="rmod_4.html#E13:70_1_1"><i><font color="Green">E58</font></i></a>, , , <a class="ref" href="rmod_4.html#T5" target="_self">Th5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="vectsp_2.html#K6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="rmod_4.html#K7" target="_self">*</a> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rmod_4.html#D4" target="_self">Def4</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="vectsp_2.html#K6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="vectsp_2.html#K6">*</a> <span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span></span>)</span></span>)</span> <a href="vectsp_2.html#K6">*</a> <font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="vectsp_2.html#D23">VECTSP_2:def 23</a></i>
<br/>.= 
<font color="Maroon">v</font> <a href="vectsp_2.html#K6">*</a> <font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="rmod_4.html#D3" target="_self">Def3</a>, , <a class="ref" href="rmod_4.html#T13" target="_self">Th13</a>, </i>
;<br/></div><b>end;</b></div><b>hence </b><a NAME="E35:70_1_1"/>
 <a href="rmod_4.html#K5" target="_self">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="rmod_4.html#K7" target="_self">*</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rmod_4.html#K5" target="_self">Sum</a> <font color="Maroon">L</font></span>)</span> <a href="vectsp_2.html#K6">*</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="txt" href="rmod_4.html#E1"><i><font color="Green">Lemma44</font></i></a>, , , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_2"><b>suppose </b></a><a NAME="E1:70_1_2"/><i><font color="Green">E127</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font>
 ;<br/><div class="add"><b>hence </b> <a href="rmod_4.html#K5" target="_self">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="rmod_4.html#K7" target="_self">*</a> <font color="Maroon">a</font></span>)</span> = 
 <a href="rmod_4.html#K5" target="_self">Sum</a> <span class="p1">(<span class="default"><a href="rmod_4.html#K3" target="_self">ZeroLC</a> <font color="Maroon">V</font></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="rmod_4.html#K5" target="_self">Sum</a> <font color="Maroon">L</font></span>)</span> <a href="vectsp_2.html#K6">*</a> <font color="Maroon">a</font>
<b>by </b><i>, <a class="ref" href="mod_1.html#T37">MOD_1:37</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
