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

<b>consider </b><font color="Maroon">c<sub>1</sub></font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">c<sub>2</sub></font> being   <a href="binop_1.html#NM2">BinOp</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>3</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E2:4_1_1"/><i><font color="Green">E6</font></i>: 
( ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>2</sub></font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span>)</span>,<font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="binop_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_1.html#K2">.</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span>)</span> ) &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>( <font color="Maroon">c<sub>2</sub></font> <a href="binop_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> ) ) &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>ex <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>2</sub></font> <a href="binop_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="binop_1.html#K2">.</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> ) ) &amp; ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>2</sub></font> <a href="binop_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="binop_1.html#K2">.</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="txt" href="realset2.html#E5"><i><font color="Green">Lemma5</font></i></a></i>;<br/>
<a NAME="E3:4_1_1"/>
(  <a href="rlvect_1.html#G1">LoopStr</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> #) is <a href="rlvect_1.html#V3">Abelian</a> &amp;  <a href="rlvect_1.html#G1">LoopStr</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> #) is <a href="rlvect_1.html#V4">add-associative</a> &amp;  <a href="rlvect_1.html#G1">LoopStr</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> #) is <a href="realset2.html#V1" target="_self">zeroed</a> &amp;  <a href="rlvect_1.html#G1">LoopStr</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> #) is <a href="realset2.html#V2" target="_self">complementable</a> )
 
<b>by </b><i><a class="txt" href="realset2.html#E2:4_1_1"><i><font color="Green">E6</font></i></a>, <a class="ref" href="realset2.html#D1" target="_self">Def1</a>, <a class="ref" href="realset2.html#D2" target="_self">Def2</a>, <a class="ref" href="realset2.html#D3" target="_self">Def3</a>, <a class="ref" href="realset2.html#D4" target="_self">Def4</a></i>;<br/>
<b>hence </b><a NAME="E4:4_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being non <a href="struct_0.html#V3">empty</a>  <a href="rlvect_1.html#L1">LoopStr</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="rlvect_1.html#V1">strict</a> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="rlvect_1.html#V3">Abelian</a> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="rlvect_1.html#V4">add-associative</a> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="realset2.html#V1" target="_self">zeroed</a> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="realset2.html#V2" target="_self">complementable</a> )
 ;<br/>


</div>
