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

<span class="kw">reconsider </span><font color="Maroon" title="c2">GS</font> =  <a href="algstr_0.html#G2" title="ALGSTR_0:aggr.2">addLoopStr</a>(# <span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p2">(<span class="default"><a href="prvect_1.html#K16" title="PRVECT_1:func.16">carr</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span>,<span class="p1"><a href="prvect_1.html#K14" title="PRVECT_1:func.14">[:</a><span class="default"><span class="p2">(<span class="default"><a href="prvect_1.html#K18" title="PRVECT_1:func.18">addop</a> <font color="Maroon" title="c1">G</font></span>)</span></span><a href="prvect_1.html#K14" title="PRVECT_1:func.14">:]</a></span>,<span class="p1">(<span class="default"><a href="prvect_1.html#K20" title="PRVECT_1:func.20">zeros</a> <font color="Maroon" title="c1">G</font></span>)</span> #) as  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="algstr_0.html#L2" title="ALGSTR_0:struct.2">addLoopStr</a>  ;<br/>
<div><span class="lab"><font color="Green" title="E36">A1</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="prvect_1.html#K16" title="PRVECT_1:func.16">carr</a> <font color="Maroon" title="c1">G</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="prvect_1.html#K16" title="PRVECT_1:func.16">carr</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">G</font> <a href="prvect_1.html#K17" title="PRVECT_1:func.17">.</a> <font color="Maroon" title="c3">i</font>)</span><br/> <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">G</font> = 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">G</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
<br/>.= 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="prvect_1.html#K16" title="PRVECT_1:func.16">carr</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="prvect_1.html#D12" target="_self" title="PRVECT_1:def.12">Def12</a></span>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="prvect_1.html#K16" title="PRVECT_1:func.16">carr</a> <font color="Maroon" title="c1">G</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
;<br/><span class="kw">hence </span><a NAME="E2:63_1_1_1"/>
<span class="p1">(<span class="default"><a href="prvect_1.html#K16" title="PRVECT_1:func.16">carr</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">G</font> <a href="prvect_1.html#K17" title="PRVECT_1:func.17">.</a> <font color="Maroon" title="c3">i</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prvect_1.html#D12" target="_self" title="PRVECT_1:def.12">Def12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="prvect_1.html#K16" title="PRVECT_1:func.16">carr</a> <font color="Maroon" title="c1">G</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="prvect_1.html#K18" title="PRVECT_1:func.18">addop</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="prvect_1.html#K10" title="PRVECT_1:func.10">.</a> <font color="Maroon" title="c3">i</font> is  <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a> </span><br/><a NAME="E1:63_1_1_2"/>
( <span class="p1">(<span class="default"><a href="prvect_1.html#K18" title="PRVECT_1:func.18">addop</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="prvect_1.html#K10" title="PRVECT_1:func.10">.</a> <font color="Maroon" title="c3">i</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">G</font> <a href="prvect_1.html#K17" title="PRVECT_1:func.17">.</a> <font color="Maroon" title="c3">i</font>) &amp; <span class="p1">(<span class="default"><a href="prvect_1.html#K16" title="PRVECT_1:func.16">carr</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">G</font> <a href="prvect_1.html#K17" title="PRVECT_1:func.17">.</a> <font color="Maroon" title="c3">i</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:63_1_1"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="ref" href="prvect_1.html#D13" target="_self" title="PRVECT_1:def.13">Def13</a></span>;<br/><span class="kw">hence </span><a NAME="E2:63_1_1_2"/>
<span class="p1">(<span class="default"><a href="prvect_1.html#K18" title="PRVECT_1:func.18">addop</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="prvect_1.html#K10" title="PRVECT_1:func.10">.</a> <font color="Maroon" title="c3">i</font> is  <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fvsum_1.html#T3" title="FVSUM_1:th.3">FVSUM_1:3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E4:63_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A2</font></span>: 
<span class="p1"><a href="prvect_1.html#K14" title="PRVECT_1:func.14">[:</a><span class="default"><span class="p2">(<span class="default"><a href="prvect_1.html#K18" title="PRVECT_1:func.18">addop</a> <font color="Maroon" title="c1">G</font></span>)</span></span><a href="prvect_1.html#K14" title="PRVECT_1:func.14">:]</a></span> is  <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="prvect_1.html#T26" target="_self" title="PRVECT_1:th.26">Th26</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1_1_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="prvect_1.html#K16" title="PRVECT_1:func.16">carr</a> <font color="Maroon" title="c1">G</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="prvect_1.html#K20" title="PRVECT_1:func.20">zeros</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="prvect_1.html#K9" title="PRVECT_1:func.9">.</a> <font color="Maroon" title="c3">i</font> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a> <span class="p1">(<span class="default"><a href="prvect_1.html#K18" title="PRVECT_1:func.18">addop</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="prvect_1.html#K10" title="PRVECT_1:func.10">.</a> <font color="Maroon" title="c3">i</font></span><br/><a NAME="E1:63_1_1_3"/><span class="lab"><font color="Green" title="E38">A3</font></span>: 
<span class="p1">(<span class="default"><a href="prvect_1.html#K16" title="PRVECT_1:func.16">carr</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">G</font> <a href="prvect_1.html#K17" title="PRVECT_1:func.17">.</a> <font color="Maroon" title="c3">i</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:63_1_1"><span class="lab"><font color="Green" title="E36">A1</font></span></a></span>;<br/><a NAME="E2:63_1_1_3"/>
( <span class="p1">(<span class="default"><a href="prvect_1.html#K18" title="PRVECT_1:func.18">addop</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="prvect_1.html#K10" title="PRVECT_1:func.10">.</a> <font color="Maroon" title="c3">i</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">G</font> <a href="prvect_1.html#K17" title="PRVECT_1:func.17">.</a> <font color="Maroon" title="c3">i</font>) &amp; <span class="p1">(<span class="default"><a href="prvect_1.html#K20" title="PRVECT_1:func.20">zeros</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="prvect_1.html#K9" title="PRVECT_1:func.9">.</a> <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon" title="c1">G</font> <a href="prvect_1.html#K17" title="PRVECT_1:func.17">.</a> <font color="Maroon" title="c3">i</font>) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prvect_1.html#D13" target="_self" title="PRVECT_1:def.13">Def13</a>, <a class="ref" href="prvect_1.html#D15" target="_self" title="PRVECT_1:def.15">Def15</a></span>;<br/><span class="kw">hence </span><a NAME="E3:63_1_1_3"/>
<span class="p1">(<span class="default"><a href="prvect_1.html#K20" title="PRVECT_1:func.20">zeros</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="prvect_1.html#K9" title="PRVECT_1:func.9">.</a> <font color="Maroon" title="c3">i</font> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a> <span class="p1">(<span class="default"><a href="prvect_1.html#K18" title="PRVECT_1:func.18">addop</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="prvect_1.html#K10" title="PRVECT_1:func.10">.</a> <font color="Maroon" title="c3">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:63_1_1_3"><span class="lab"><font color="Green" title="E38">A3</font></span></a>, <a class="ref" href="prvect_1.html#T3" target="_self" title="PRVECT_1:th.3">Th3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E6:63_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A4</font></span>: 
 <a href="prvect_1.html#K20" title="PRVECT_1:func.20">zeros</a> <font color="Maroon" title="c1">G</font> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a> <span class="p1"><a href="prvect_1.html#K14" title="PRVECT_1:func.14">[:</a><span class="default"><span class="p2">(<span class="default"><a href="prvect_1.html#K18" title="PRVECT_1:func.18">addop</a> <font color="Maroon" title="c1">G</font></span>)</span></span><a href="prvect_1.html#K14" title="PRVECT_1:func.14">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="prvect_1.html#T27" target="_self" title="PRVECT_1:th.27">Th27</a></span>;<br/>
<a NAME="E7:63_1_1"/><span class="lab"><font color="Green" title="E39">A5</font></span>: 
<font color="Maroon" title="c2">GS</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="63_1_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">GS</font>; <span class="comment"><font color="Red">:: according to </font></span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">x</font> is  <a href="algstr_0.html#V10" title="ALGSTR_0:attr.10">right_complementable</a> </span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c4">y</font> = <span class="p1">(<span class="default"><a href="prvect_1.html#K12" title="PRVECT_1:func.12">Frege</a> <span class="p2">(<span class="default"><a href="prvect_1.html#K19" title="PRVECT_1:func.19">complop</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">GS</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c4">y</font>
; <span class="comment"><font color="Red">:: according to </font></span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c4">y</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">GS</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1_1_4_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="prvect_1.html#K16" title="PRVECT_1:func.16">carr</a> <font color="Maroon" title="c1">G</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="prvect_1.html#K19" title="PRVECT_1:func.19">complop</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="prvect_1.html#K11" title="PRVECT_1:func.11">.</a> <font color="Maroon" title="c5">i</font> <a href="finseqop.html#R1" title="FINSEQOP:pred.1">is_an_inverseOp_wrt</a> <span class="p1">(<span class="default"><a href="prvect_1.html#K18" title="PRVECT_1:func.18">addop</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="prvect_1.html#K10" title="PRVECT_1:func.10">.</a> <font color="Maroon" title="c5">i</font> &amp; <span class="p1">(<span class="default"><a href="prvect_1.html#K18" title="PRVECT_1:func.18">addop</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="prvect_1.html#K10" title="PRVECT_1:func.10">.</a> <font color="Maroon" title="c5">i</font> is  <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_a_unity</a>  )</span><br/><a NAME="E1:63_1_1_4_1"/><span class="lab"><font color="Green" title="E40">A6</font></span>: 
( <span class="p1">(<span class="default"><a href="prvect_1.html#K18" title="PRVECT_1:func.18">addop</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="prvect_1.html#K10" title="PRVECT_1:func.10">.</a> <font color="Maroon" title="c5">i</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">G</font> <a href="prvect_1.html#K17" title="PRVECT_1:func.17">.</a> <font color="Maroon" title="c5">i</font>) &amp; <span class="p1">(<span class="default"><a href="prvect_1.html#K19" title="PRVECT_1:func.19">complop</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="prvect_1.html#K11" title="PRVECT_1:func.11">.</a> <font color="Maroon" title="c5">i</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c1">G</font> <a href="prvect_1.html#K17" title="PRVECT_1:func.17">.</a> <font color="Maroon" title="c5">i</font>) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prvect_1.html#D13" target="_self" title="PRVECT_1:def.13">Def13</a>, <a class="ref" href="prvect_1.html#D14" target="_self" title="PRVECT_1:def.14">Def14</a></span>;<br/><a NAME="E2:63_1_1_4_1"/>
( <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon" title="c1">G</font> <a href="prvect_1.html#K17" title="PRVECT_1:func.17">.</a> <font color="Maroon" title="c5">i</font>) <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">G</font> <a href="prvect_1.html#K17" title="PRVECT_1:func.17">.</a> <font color="Maroon" title="c5">i</font>) &amp; <span class="p1">(<span class="default"><a href="prvect_1.html#K16" title="PRVECT_1:func.16">carr</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">G</font> <a href="prvect_1.html#K17" title="PRVECT_1:func.17">.</a> <font color="Maroon" title="c5">i</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:63_1_1"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="ref" href="prvect_1.html#T3" target="_self" title="PRVECT_1:th.3">Th3</a></span>;<br/><span class="kw">hence </span><a NAME="E3:63_1_1_4_1"/>
( <span class="p1">(<span class="default"><a href="prvect_1.html#K19" title="PRVECT_1:func.19">complop</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="prvect_1.html#K11" title="PRVECT_1:func.11">.</a> <font color="Maroon" title="c5">i</font> <a href="finseqop.html#R1" title="FINSEQOP:pred.1">is_an_inverseOp_wrt</a> <span class="p1">(<span class="default"><a href="prvect_1.html#K18" title="PRVECT_1:func.18">addop</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="prvect_1.html#K10" title="PRVECT_1:func.10">.</a> <font color="Maroon" title="c5">i</font> &amp; <span class="p1">(<span class="default"><a href="prvect_1.html#K18" title="PRVECT_1:func.18">addop</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="prvect_1.html#K10" title="PRVECT_1:func.10">.</a> <font color="Maroon" title="c5">i</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="txt" href="#E1:63_1_1_4_1"><span class="lab"><font color="Green" title="E40">A6</font></span></a>, <a class="ref" href="prvect_1.html#T4" target="_self" title="PRVECT_1:th.4">Th4</a>, <a class="ref" href="setwiseo.html#D2" title="SETWISEO:def.2">SETWISEO:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E3:63_1_1_4"/><span class="kw">then </span>
 <a href="prvect_1.html#K12" title="PRVECT_1:func.12">Frege</a> <span class="p1">(<span class="default"><a href="prvect_1.html#K19" title="PRVECT_1:func.19">complop</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="finseqop.html#R1" title="FINSEQOP:pred.1">is_an_inverseOp_wrt</a> <span class="p1"><a href="prvect_1.html#K14" title="PRVECT_1:func.14">[:</a><span class="default"><span class="p2">(<span class="default"><a href="prvect_1.html#K18" title="PRVECT_1:func.18">addop</a> <font color="Maroon" title="c1">G</font></span>)</span></span><a href="prvect_1.html#K14" title="PRVECT_1:func.14">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="prvect_1.html#T28" target="_self" title="PRVECT_1:th.28">Th28</a></span>;<br/>
<a NAME="E4:63_1_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <span class="p1"><a href="prvect_1.html#K14" title="PRVECT_1:func.14">[:</a><span class="default"><span class="p2">(<span class="default"><a href="prvect_1.html#K18" title="PRVECT_1:func.18">addop</a> <font color="Maroon" title="c1">G</font></span>)</span></span><a href="prvect_1.html#K14" title="PRVECT_1:func.14">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseqop.html#D1" title="FINSEQOP:def.1">FINSEQOP:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:63_1_1_4"/>
<font color="Maroon" title="c3">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c4">y</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">GS</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:63_1_1"><span class="lab"><font color="Green" title="E38">A4</font></span></a>, <a class="ref" href="binop_1.html#D8" title="BINOP_1:def.8">BINOP_1:def 8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1_1_5"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="prvect_1.html#K16" title="PRVECT_1:func.16">carr</a> <font color="Maroon" title="c1">G</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="prvect_1.html#K18" title="PRVECT_1:func.18">addop</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="prvect_1.html#K10" title="PRVECT_1:func.10">.</a> <font color="Maroon" title="c3">i</font> is  <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a> </span><br/><a NAME="E1:63_1_1_5"/>
( <span class="p1">(<span class="default"><a href="prvect_1.html#K18" title="PRVECT_1:func.18">addop</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="prvect_1.html#K10" title="PRVECT_1:func.10">.</a> <font color="Maroon" title="c3">i</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">G</font> <a href="prvect_1.html#K17" title="PRVECT_1:func.17">.</a> <font color="Maroon" title="c3">i</font>) &amp; <span class="p1">(<span class="default"><a href="prvect_1.html#K16" title="PRVECT_1:func.16">carr</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">G</font> <a href="prvect_1.html#K17" title="PRVECT_1:func.17">.</a> <font color="Maroon" title="c3">i</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:63_1_1"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="ref" href="prvect_1.html#D13" target="_self" title="PRVECT_1:def.13">Def13</a></span>;<br/><span class="kw">hence </span><a NAME="E2:63_1_1_5"/>
<span class="p1">(<span class="default"><a href="prvect_1.html#K18" title="PRVECT_1:func.18">addop</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="prvect_1.html#K10" title="PRVECT_1:func.10">.</a> <font color="Maroon" title="c3">i</font> is  <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fvsum_1.html#T2" title="FVSUM_1:th.2">FVSUM_1:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E9:63_1_1"/><span class="kw">then </span>
(  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">GS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="prvect_1.html#K20" title="PRVECT_1:func.20">zeros</a> <font color="Maroon" title="c1">G</font> &amp; <span class="p1"><a href="prvect_1.html#K14" title="PRVECT_1:func.14">[:</a><span class="default"><span class="p2">(<span class="default"><a href="prvect_1.html#K18" title="PRVECT_1:func.18">addop</a> <font color="Maroon" title="c1">G</font></span>)</span></span><a href="prvect_1.html#K14" title="PRVECT_1:func.14">:]</a></span> is  <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="prvect_1.html#T25" target="_self" title="PRVECT_1:th.25">Th25</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:63_1_1"/>
 <a href="algstr_0.html#G2" title="ALGSTR_0:aggr.2">addLoopStr</a>(# <span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p2">(<span class="default"><a href="prvect_1.html#K16" title="PRVECT_1:func.16">carr</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span>,<span class="p1"><a href="prvect_1.html#K14" title="PRVECT_1:func.14">[:</a><span class="default"><span class="p2">(<span class="default"><a href="prvect_1.html#K18" title="PRVECT_1:func.18">addop</a> <font color="Maroon" title="c1">G</font></span>)</span></span><a href="prvect_1.html#K14" title="PRVECT_1:func.14">:]</a></span>,<span class="p1">(<span class="default"><a href="prvect_1.html#K20" title="PRVECT_1:func.20">zeros</a> <font color="Maroon" title="c1">G</font></span>)</span> #) is   <a href="algstr_0.html#V8" title="ALGSTR_0:attr.8">strict</a>  <a href="vectsp_1.html#NM2" title="VECTSP_1:NM.2">AbGroup</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:63_1_1"><span class="lab"><font color="Green" title="E37">A2</font></span></a>, <a class="txt" href="#E6:63_1_1"><span class="lab"><font color="Green" title="E38">A4</font></span></a>, <a class="txt" href="#E7:63_1_1"><span class="lab"><font color="Green" title="E39">A5</font></span></a>, <a class="txt" href="prvect_1.html#E7"><span class="lab"><font color="Green" title="E4">Lm2</font></span></a>, <a class="txt" href="prvect_1.html#E8"><span class="lab"><font color="Green" title="E5">Lm3</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
