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

<b>set </b><font color="Maroon" title="c12">L</font> =  <a href="algstr_0.html#G6" title="ALGSTR_0:aggr.6">doubleLoopStr</a>(# <font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c8">od0</font>,<font color="Maroon" title="c9">om0</font>,<font color="Maroon" title="c7">nm</font>,<font color="Maroon" title="c5">nd</font> #);<br/>
<a NAME="E1:25_1_1"/><i><font color="Green" title="E27">A1</font></i>: 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="algstr_0.html#G6" title="ALGSTR_0:aggr.6">doubleLoopStr</a>(# <font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c8">od0</font>,<font color="Maroon" title="c9">om0</font>,<font color="Maroon" title="c7">nm</font>,<font color="Maroon" title="c5">nd</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">nd</font>
 
;<br/>
<a NAME="E2:25_1_1"/><i><font color="Green" title="E28">A2</font></i>: 
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <a href="algstr_0.html#G6" title="ALGSTR_0:aggr.6">doubleLoopStr</a>(# <font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c8">od0</font>,<font color="Maroon" title="c9">om0</font>,<font color="Maroon" title="c7">nm</font>,<font color="Maroon" title="c5">nd</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">nm</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c13">L</font> =  <a href="algstr_0.html#G6" title="ALGSTR_0:aggr.6">doubleLoopStr</a>(# <font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c8">od0</font>,<font color="Maroon" title="c9">om0</font>,<font color="Maroon" title="c7">nm</font>,<font color="Maroon" title="c5">nd</font> #) as  non <a href="struct_0.html#V6" title="STRUCT_0:attr.6">degenerated</a>  <a href="algstr_0.html#L6" title="ALGSTR_0:struct.6">doubleLoopStr</a>  <b>by </b><i><a class="txt" href="realset2.html#E1:25_1_1"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="realset2.html#E11"><i><font color="Green" title="E2">Lm2</font></i></a>, <a class="ref" href="struct_0.html#D8" title="STRUCT_0:def.8">STRUCT_0:def 8</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c13">L</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c13">L</font> is <a href="algstr_0.html#V36" title="ALGSTR_0:attr.36">strict</a> &amp; <font color="Maroon" title="c13">L</font> is <a href="realset2.html#V1" title="REALSET2:attr.1">Field-like</a> &amp; <font color="Maroon" title="c13">L</font> is <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a> &amp; <font color="Maroon" title="c13">L</font> is <a href="vectsp_1.html#V5" title="VECTSP_1:attr.5">distributive</a> &amp; <font color="Maroon" title="c13">L</font> is <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a> &amp; <font color="Maroon" title="c13">L</font> is <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a> &amp; <font color="Maroon" title="c13">L</font> is <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a> )</span><br/>

<a NAME="E4:25_1_1"/>
<font color="Maroon" title="c13">L</font> is <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c13">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="algstr_0.html#D16" title="ALGSTR_0:def.16">ALGSTR_0:def 16</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c14">a</font> is <a href="algstr_0.html#V10" title="ALGSTR_0:attr.10">right_complementable</a></span><br/>

<b>take </b>
<font color="Maroon" title="c14">a</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="algstr_0.html#D11" title="ALGSTR_0:def.11">ALGSTR_0:def 11</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c14">a</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c14">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c13">L</font></span><br/>

<a NAME="E1:25_1_1_1"/>
( <font color="Maroon" title="c14">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,2 or <font color="Maroon" title="c14">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> 1,2 )
 
<b>by </b><i><a class="txt" href="realset2.html#E11"><i><font color="Green" title="E2">Lm2</font></i></a>, <a class="ref" href="card_1.html#T88" title="CARD_1:th.88">CARD_1:88</a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>
<b>hence </b><a NAME="E2:25_1_1_1"/>
<font color="Maroon" title="c14">a</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c14">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c13">L</font>
 <b>by </b><i><a class="txt" href="realset2.html#E17"><i><font color="Green" title="E6">Lm6</font></i></a>, <a class="txt" href="realset2.html#E20"><i><font color="Green" title="E9">Lm9</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:25_1_1"/>
( <font color="Maroon" title="c13">L</font> is <a href="algstr_0.html#V36" title="ALGSTR_0:attr.36">strict</a> &amp; <font color="Maroon" title="c13">L</font> is <a href="realset2.html#V1" title="REALSET2:attr.1">Field-like</a> &amp; <font color="Maroon" title="c13">L</font> is <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a> &amp; <font color="Maroon" title="c13">L</font> is <a href="vectsp_1.html#V5" title="VECTSP_1:attr.5">distributive</a> &amp; <font color="Maroon" title="c13">L</font> is <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a> &amp; <font color="Maroon" title="c13">L</font> is <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a> &amp; <font color="Maroon" title="c13">L</font> is <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a> )
 <b>by </b><i><a class="txt" href="realset2.html#E2:25_1_1"><i><font color="Green" title="E28">A2</font></i></a>, <a class="ref" href="realset2.html#D11" target="_self" title="REALSET2:def.11">Def11</a>, <a class="txt" href="realset2.html#E36"><i><font color="Green" title="E21">Lm21</font></i></a>, <a class="txt" href="realset2.html#E37"><i><font color="Green" title="E22">Lm22</font></i></a>, <a class="txt" href="realset2.html#E38"><i><font color="Green" title="E23">Lm23</font></i></a>, <a class="txt" href="realset2.html#E40"><i><font color="Green" title="E24">Lm24</font></i></a>, <a class="txt" href="realset2.html#E41"><i><font color="Green" title="E25">Lm25</font></i></a>, <a class="ref" href="rlvect_1.html#D5" title="RLVECT_1:def.5">RLVECT_1:def 5</a>, <a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a>, <a class="ref" href="rlvect_1.html#D7" title="RLVECT_1:def.7">RLVECT_1:def 7</a>, <a class="ref" href="vectsp_1.html#D18" title="VECTSP_1:def.18">VECTSP_1:def 18</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
