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<b>let </b><font color="Maroon" title="c1">V</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">Abelian</a> <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">add-associative</a> <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">right_zeroed</a> <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">right_complementable</a>  <a href="rlvect_1.html#L1" title="RLVECT_1:struct.1">LoopStr</a> ;<br/><b>let </b><font color="Maroon" title="c2">v1</font>, <font color="Maroon" title="c3">v2</font>, <font color="Maroon" title="c4">v3</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font>;<br/>



<b>assume </b><a NAME="E1:15"/>
( <font color="Maroon" title="c2">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">v2</font> &amp; <font color="Maroon" title="c3">v2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">v3</font> &amp; <font color="Maroon" title="c2">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">v3</font> )
 ;<br/>

<a NAME="E2:15"/><b>then </b>
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c3">v2</font>,<font color="Maroon" title="c4">v3</font></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c3">v2</font>,<font color="Maroon" title="c4">v3</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> &amp; <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c3">v2</font>,<font color="Maroon" title="c4">v3</font></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp;  <a href="rlvect_1.html#K9" title="RLVECT_1:func.9">Sum</a> <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c3">v2</font>,<font color="Maroon" title="c4">v3</font></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">v1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c3">v2</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c4">v3</font> )
 
<b>by </b><i><a class="ref" href="finseq_2.html#T148" title="FINSEQ_2:th.148">FINSEQ_2:148</a>, <a class="ref" href="finseq_3.html#T104" title="FINSEQ_3:th.104">FINSEQ_3:104</a>, <a class="ref" href="rlvect_1.html#T63" title="RLVECT_1:th.63">RLVECT_1:63</a></i>;<br/>
<b>hence </b><a NAME="E3:15"/>
 <a href="rlvect_2.html#K2" title="RLVECT_2:func.2">Sum</a> <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c3">v2</font>,<font color="Maroon" title="c4">v3</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">v1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c3">v2</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c4">v3</font>
 <b>by </b><i><a class="ref" href="rlvect_2.html#D4" target="_self" title="RLVECT_2:def.4">Def4</a></i>;<br/>


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