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

<b>let </b><font color="Maroon">GF</font> be  non <a href="struct_0.html#V3">empty</a> <a href="rlvect_1.html#V3">Abelian</a> <a href="rlvect_1.html#V4">add-associative</a> <a href="rlvect_1.html#V5">right_zeroed</a> <a href="rlvect_1.html#V6">right_complementable</a> <a href="group_1.html#V4">associative</a> <a href="vectsp_1.html#V7">distributive</a> <a href="vectsp_1.html#V8">left_unital</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon">V</font> be  non <a href="struct_0.html#V3">empty</a> <a href="rlvect_1.html#V3">Abelian</a> <a href="rlvect_1.html#V4">add-associative</a> <a href="rlvect_1.html#V5">right_zeroed</a> <a href="rlvect_1.html#V6">right_complementable</a> <a href="vectsp_1.html#V12">VectSp-like</a>  <a href="vectsp_1.html#L4">VectSpStr</a> of <font color="Maroon">GF</font>;<br/><b>let </b><font color="Maroon">v1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">v2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">v3</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">GF</font>;<br/>











<i><font color="Green">E47</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="vectsp_6.html#K3" target="_self">(#)</a> <span class="p2"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><font color="Maroon">v1</font>,<font color="Maroon">v2</font>,<font color="Maroon">v3</font></span><a href="finseq_4.html#K3">*&gt;</a></span></span>)</span> = 
 <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><font color="Maroon">v1</font>,<font color="Maroon">v2</font>,<font color="Maroon">v3</font></span><a href="finseq_4.html#K3">*&gt;</a></span>
<b>by </b><i/>
<br/>.= 
3
<b>by </b><i><a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>
;<br/>
<a NAME="E2:15"/><b>then </b><i><font color="Green">E55</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="vectsp_6.html#K3" target="_self">(#)</a> <span class="p2"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><font color="Maroon">v1</font>,<font color="Maroon">v2</font>,<font color="Maroon">v3</font></span><a href="finseq_4.html#K3">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K8">{</a><span class="default">1,2,3</span><a href="domain_1.html#K8">}</a></span> &amp; 1 <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K8">{</a><span class="default">1,2,3</span><a href="domain_1.html#K8">}</a></span> &amp; 2 <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K8">{</a><span class="default">1,2,3</span><a href="domain_1.html#K8">}</a></span> &amp; 3 <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K8">{</a><span class="default">1,2,3</span><a href="domain_1.html#K8">}</a></span> )
 
<b>by </b><i><a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a>, <a class="ref" href="finseq_3.html#T1">FINSEQ_3:1</a></i>;<br/>
<b>then </b><i><font color="Green">E59</font></i>: <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="vectsp_6.html#K3" target="_self">(#)</a> <span class="p2"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><font color="Maroon">v1</font>,<font color="Maroon">v2</font>,<font color="Maroon">v3</font></span><a href="finseq_4.html#K3">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><span class="p3"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><font color="Maroon">v1</font>,<font color="Maroon">v2</font>,<font color="Maroon">v3</font></span><a href="finseq_4.html#K3">*&gt;</a></span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span> <a href="vectsp_1.html#K6">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><font color="Maroon">v1</font>,<font color="Maroon">v2</font>,<font color="Maroon">v3</font></span><a href="finseq_4.html#K3">*&gt;</a></span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><span class="p3"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><font color="Maroon">v1</font>,<font color="Maroon">v2</font>,<font color="Maroon">v3</font></span><a href="finseq_4.html#K3">*&gt;</a></span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v1</font>
<b>by </b><i><a class="ref" href="finseq_4.html#T27">FINSEQ_4:27</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v1</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v1</font>
<b>by </b><i><a class="ref" href="finseq_4.html#T27">FINSEQ_4:27</a></i>
;<br/>
<i><font color="Green">E60</font></i>: <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="vectsp_6.html#K3" target="_self">(#)</a> <span class="p2"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><font color="Maroon">v1</font>,<font color="Maroon">v2</font>,<font color="Maroon">v3</font></span><a href="finseq_4.html#K3">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> 2 = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><span class="p3"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><font color="Maroon">v1</font>,<font color="Maroon">v2</font>,<font color="Maroon">v3</font></span><a href="finseq_4.html#K3">*&gt;</a></span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span> <a href="vectsp_1.html#K6">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><font color="Maroon">v1</font>,<font color="Maroon">v2</font>,<font color="Maroon">v3</font></span><a href="finseq_4.html#K3">*&gt;</a></span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><span class="p3"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><font color="Maroon">v1</font>,<font color="Maroon">v2</font>,<font color="Maroon">v3</font></span><a href="finseq_4.html#K3">*&gt;</a></span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v2</font>
<b>by </b><i><a class="ref" href="finseq_4.html#T27">FINSEQ_4:27</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v2</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v2</font>
<b>by </b><i><a class="ref" href="finseq_4.html#T27">FINSEQ_4:27</a></i>
;<br/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="vectsp_6.html#K3" target="_self">(#)</a> <span class="p2"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><font color="Maroon">v1</font>,<font color="Maroon">v2</font>,<font color="Maroon">v3</font></span><a href="finseq_4.html#K3">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> 3 = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><span class="p3"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><font color="Maroon">v1</font>,<font color="Maroon">v2</font>,<font color="Maroon">v3</font></span><a href="finseq_4.html#K3">*&gt;</a></span> <a href="finseq_4.html#K4">/.</a> 3</span>)</span></span>)</span> <a href="vectsp_1.html#K6">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><font color="Maroon">v1</font>,<font color="Maroon">v2</font>,<font color="Maroon">v3</font></span><a href="finseq_4.html#K3">*&gt;</a></span> <a href="finseq_4.html#K4">/.</a> 3</span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><span class="p3"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><font color="Maroon">v1</font>,<font color="Maroon">v2</font>,<font color="Maroon">v3</font></span><a href="finseq_4.html#K3">*&gt;</a></span> <a href="finseq_4.html#K4">/.</a> 3</span>)</span></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v3</font>
<b>by </b><i><a class="ref" href="finseq_4.html#T27">FINSEQ_4:27</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v3</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v3</font>
<b>by </b><i><a class="ref" href="finseq_4.html#T27">FINSEQ_4:27</a></i>
;<br/>
<b>hence </b><a NAME="E6:15"/>
<font color="Maroon">f</font> <a href="vectsp_6.html#K3" target="_self">(#)</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><font color="Maroon">v1</font>,<font color="Maroon">v2</font>,<font color="Maroon">v3</font></span><a href="finseq_4.html#K3">*&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v1</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v1</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v2</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v3</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v3</font></span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span>
 <b>by </b><i>, , , <a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>;<br/>


</div>
