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

<span class="kw">let </span><font color="Maroon" title="c1">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">K</font> being   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>   <a href="algstr_1.html#V1" title="ALGSTR_1:attr.1">left_zeroed</a>   <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>   <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>   <a href="algstr_0.html#L2" title="ALGSTR_0:struct.2">addLoopStr</a> <br/>  for <font color="Olive" title="b2">R1</font>, <font color="Olive" title="b3">R2</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">i</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">K</font>  st <font color="Olive" title="b2">R1</font> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <font color="Olive" title="b3">R2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">i</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Olive" title="b1">K</font></span>)</span> holds <br/>( <font color="Olive" title="b2">R1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fvsum_1.html#K6" title="FVSUM_1:func.6">-</a> <font color="Olive" title="b3">R2</font> &amp; <font color="Olive" title="b3">R2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fvsum_1.html#K6" title="FVSUM_1:func.6">-</a> <font color="Olive" title="b2">R1</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">K</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>   <a href="algstr_1.html#V1" title="ALGSTR_1:attr.1">left_zeroed</a>   <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>   <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>   <a href="algstr_0.html#L2" title="ALGSTR_0:struct.2">addLoopStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">R1</font>, <font color="Olive" title="b2">R2</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">i</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">K</font>  st <font color="Olive" title="b1">R1</font> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <font color="Olive" title="b2">R2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">i</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span> holds <br/>( <font color="Olive" title="b1">R1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fvsum_1.html#K6" title="FVSUM_1:func.6">-</a> <font color="Olive" title="b2">R2</font> &amp; <font color="Olive" title="b2">R2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fvsum_1.html#K6" title="FVSUM_1:func.6">-</a> <font color="Olive" title="b1">R1</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">R1</font>, <font color="Maroon" title="c4">R2</font> be    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">i</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">K</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">R1</font> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <font color="Maroon" title="c4">R2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">i</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span> implies ( <font color="Maroon" title="c3">R1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fvsum_1.html#K6" title="FVSUM_1:func.6">-</a> <font color="Maroon" title="c4">R2</font> &amp; <font color="Maroon" title="c4">R2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fvsum_1.html#K6" title="FVSUM_1:func.6">-</a> <font color="Maroon" title="c3">R1</font> ) )</span><br/>







<a NAME="E1:35"/><span class="lab"><font color="Green" title="E23">A1</font></span>: 
(  the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">K</font> is  <a href="finseqop.html#V1" title="FINSEQOP:attr.1">having_an_inverseOp</a>  &amp;  <a href="finseqop.html#K6" title="FINSEQOP:func.6">the_inverseOp_wrt</a>  the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">K</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">comp</a> <font color="Maroon" title="c2">K</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fvsum_1.html#T14" target="_self" title="FVSUM_1:th.14">Th17</a>, <a class="ref" href="fvsum_1.html#T15" target="_self" title="FVSUM_1:th.15">Th18</a></span>;<br/>
<a NAME="E2:35"/>
(  <a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a>  the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">K</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="c2">K</font> &amp;  the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">K</font> is  <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_a_unity</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fvsum_1.html#T7" target="_self" title="FVSUM_1:th.7">Th9</a>, <a class="ref" href="fvsum_1.html#T8" target="_self" title="FVSUM_1:th.8">Th10</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:35"/>
( <font color="Maroon" title="c3">R1</font> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <font color="Maroon" title="c4">R2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">i</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span> implies ( <font color="Maroon" title="c3">R1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fvsum_1.html#K6" title="FVSUM_1:func.6">-</a> <font color="Maroon" title="c4">R2</font> &amp; <font color="Maroon" title="c4">R2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fvsum_1.html#K6" title="FVSUM_1:func.6">-</a> <font color="Maroon" title="c3">R1</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="ref" href="finseqop.html#T74" title="FINSEQOP:th.74">FINSEQOP:74</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
