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<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="rlvect_1.html#L1" title="RLVECT_1:struct.1">LoopStr</a> ) <b>-&gt; </b>  <a href="relset_1.html#M2" title="RELSET_1:mode.2">M5</a>(<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>) = the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c1">G</font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="struct_0.html#L1" title="STRUCT_0:struct.1">1-sorted</a> ) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c2">GS</font> =  <a href="rlvect_1.html#G2" title="RLVECT_1:aggr.2">RLSStruct</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_2.html#K4" title="PRVECT_2:func.4">carr</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="prvect_2.html#K8" title="PRVECT_2:func.8">zeros</a> <font color="Maroon" title="c1">G</font></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_2.html#K6" title="PRVECT_2:func.6">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"><a href="prvect_2.html#K2" title="PRVECT_2:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="prvect_2.html#K9" title="PRVECT_2:func.9">multop</a> <font color="Maroon" title="c1">G</font></span>)</span></span><a href="prvect_2.html#K2" title="PRVECT_2:func.2">:]</a></span> #) as  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="rlvect_1.html#L2" title="RLVECT_1:struct.2">RLSStruct</a>  ;<br/>
<div><i><font color="Green" title="E18">A1</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_1"><b>now </b></a><div class="add"><b>let </b><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_2.html#K4" title="PRVECT_2:func.4">carr</a> <font color="Maroon" title="c1">G</font></span>)</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>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
<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_2.html#K4" title="PRVECT_2:func.4">carr</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="prvect_2.html#D4" target="_self" title="PRVECT_2:def.4">Def4</a></i>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="prvect_2.html#K4" title="PRVECT_2:func.4">carr</a> <font color="Maroon" title="c1">G</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
;<br/><b>hence </b><a NAME="E2:23_1_1_1"/>
<span class="p1">(<span class="default"><a href="prvect_2.html#K4" title="PRVECT_2:func.4">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>2</sub></font>(<font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K5" title="PRVECT_2:func.5">.</a> <font color="Maroon" title="c3">i</font>)
 <b>by </b><i><a class="ref" href="prvect_2.html#D4" target="_self" title="PRVECT_2:def.4">Def4</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_2"><b>now </b></a><div class="add"><b>let </b><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_2.html#K4" title="PRVECT_2:func.4">carr</a> <font color="Maroon" title="c1">G</font></span>)</span>;<br/><a NAME="E1:23_1_1_2"/>
( <span class="p1">(<span class="default"><a href="prvect_2.html#K6" title="PRVECT_2:func.6">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>1</sub></font>(<font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K5" title="PRVECT_2:func.5">.</a> <font color="Maroon" title="c3">i</font>) &amp; <span class="p1">(<span class="default"><a href="prvect_2.html#K4" title="PRVECT_2:func.4">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>2</sub></font>(<font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K5" title="PRVECT_2:func.5">.</a> <font color="Maroon" title="c3">i</font>) )
 <b>by </b><i><a class="txt" href="prvect_2.html#E2:23_1_1"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="prvect_2.html#D5" target="_self" title="PRVECT_2:def.5">Def5</a></i>;<br/><b>hence </b><a NAME="E2:23_1_1_2"/>
<span class="p1">(<span class="default"><a href="prvect_2.html#K6" title="PRVECT_2:func.6">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>
 <b>by </b><i><a class="ref" href="fvsum_1.html#T2" title="FVSUM_1:th.2">FVSUM_1:2</a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:23_1_1"/><b>then </b><i><font color="Green" title="E19">A2</font></i>: 
<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_2.html#K6" title="PRVECT_2:func.6">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>
 
<b>by </b><i><a class="ref" href="prvect_1.html#T25" title="PRVECT_1:th.25">PRVECT_1:25</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_3"><b>now </b></a><div class="add"><b>let </b><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_2.html#K4" title="PRVECT_2:func.4">carr</a> <font color="Maroon" title="c1">G</font></span>)</span>;<br/><a NAME="E1:23_1_1_3"/>
( <span class="p1">(<span class="default"><a href="prvect_2.html#K6" title="PRVECT_2:func.6">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>1</sub></font>(<font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K5" title="PRVECT_2:func.5">.</a> <font color="Maroon" title="c3">i</font>) &amp; <span class="p1">(<span class="default"><a href="prvect_2.html#K4" title="PRVECT_2:func.4">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>2</sub></font>(<font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K5" title="PRVECT_2:func.5">.</a> <font color="Maroon" title="c3">i</font>) )
 <b>by </b><i><a class="txt" href="prvect_2.html#E2:23_1_1"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="prvect_2.html#D5" target="_self" title="PRVECT_2:def.5">Def5</a></i>;<br/><b>hence </b><a NAME="E2:23_1_1_3"/>
<span class="p1">(<span class="default"><a href="prvect_2.html#K6" title="PRVECT_2:func.6">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>
 <b>by </b><i><a class="ref" href="fvsum_1.html#T3" title="FVSUM_1:th.3">FVSUM_1:3</a></i>;<br/></div><b>end;</b></div>
<a NAME="E6:23_1_1"/><b>then </b><i><font color="Green" title="E20">A3</font></i>: 
<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_2.html#K6" title="PRVECT_2:func.6">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>
 
<b>by </b><i><a class="ref" href="prvect_1.html#T26" title="PRVECT_1:th.26">PRVECT_1:26</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>3</sub></font>(   <a href="rlvect_1.html#L1" title="RLVECT_1:struct.1">LoopStr</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font> = the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">ZeroF</a> of <font color="Maroon" title="c1">G</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_4"><b>now </b></a><div class="add"><b>let </b><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_2.html#K4" title="PRVECT_2:func.4">carr</a> <font color="Maroon" title="c1">G</font></span>)</span>;<br/><a NAME="E1:23_1_1_4"/><i><font color="Green" title="E21">A4</font></i>: 
( <span class="p1">(<span class="default"><a href="prvect_2.html#K6" title="PRVECT_2:func.6">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>1</sub></font>(<font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K5" title="PRVECT_2:func.5">.</a> <font color="Maroon" title="c3">i</font>) &amp; <span class="p1">(<span class="default"><a href="prvect_2.html#K8" title="PRVECT_2:func.8">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>3</sub></font>(<font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K5" title="PRVECT_2:func.5">.</a> <font color="Maroon" title="c3">i</font>) &amp; <span class="p1">(<span class="default"><a href="prvect_2.html#K4" title="PRVECT_2:func.4">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>2</sub></font>(<font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K5" title="PRVECT_2:func.5">.</a> <font color="Maroon" title="c3">i</font>) )
 <b>by </b><i><a class="txt" href="prvect_2.html#E2:23_1_1"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="prvect_2.html#D5" target="_self" title="PRVECT_2:def.5">Def5</a>, <a class="ref" href="prvect_2.html#D7" target="_self" title="PRVECT_2:def.7">Def7</a></i>;<br/><a NAME="E2:23_1_1_4"/>
<span class="p1">(<span class="default"><a href="prvect_2.html#K8" title="PRVECT_2:func.8">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>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K5" title="PRVECT_2:func.5">.</a> <font color="Maroon" title="c3">i</font></span>)</span>
 <b>by </b><i><a class="ref" href="prvect_2.html#D7" target="_self" title="PRVECT_2:def.7">Def7</a></i>;<br/><b>hence </b><a NAME="E3:23_1_1_4"/>
<span class="p1">(<span class="default"><a href="prvect_2.html#K8" title="PRVECT_2:func.8">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_2.html#K6" title="PRVECT_2:func.6">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>
 <b>by </b><i><a class="txt" href="prvect_2.html#E1:23_1_1_4"><i><font color="Green" title="E21">A4</font></i></a>, <a class="ref" href="prvect_1.html#T3" title="PRVECT_1:th.3">PRVECT_1:3</a></i>;<br/></div><b>end;</b></div>
<a NAME="E8:23_1_1"/><b>then </b><i><font color="Green" title="E21">A5</font></i>: 
 <a href="prvect_2.html#K8" title="PRVECT_2:func.8">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_2.html#K6" title="PRVECT_2:func.6">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>
 
<b>by </b><i><a class="ref" href="prvect_1.html#T27" title="PRVECT_1:th.27">PRVECT_1:27</a></i>;<br/>
<a NAME="E9:23_1_1"/>
<font color="Maroon" title="c2">GS</font> is <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">right_complementable</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_5"><b>proof </b></a><div class="add">

<b>let </b><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>; <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D8" title="RLVECT_1:def.8">RLVECT_1:def 8</a><br/>

<b>reconsider </b><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_2.html#K7" title="PRVECT_2:func.7">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> <b>by </b><i><a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c4">y</font>
;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_5_1"><b>now </b></a><div class="add"><b>let </b><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_2.html#K4" title="PRVECT_2:func.4">carr</a> <font color="Maroon" title="c1">G</font></span>)</span>;<br/><a NAME="E1:23_1_1_5_1"/>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K5" title="PRVECT_2:func.5">.</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K5" title="PRVECT_2:func.5">.</a> <font color="Maroon" title="c5">i</font>)
 ;<br/><a NAME="E2:23_1_1_5_1"/><b>then </b>
( <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K5" title="PRVECT_2:func.5">.</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>1</sub></font>(<font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K5" title="PRVECT_2:func.5">.</a> <font color="Maroon" title="c5">i</font>) &amp; <span class="p1">(<span class="default"><a href="prvect_2.html#K4" title="PRVECT_2:func.4">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>2</sub></font>(<font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K5" title="PRVECT_2:func.5">.</a> <font color="Maroon" title="c5">i</font>) &amp; <span class="p1">(<span class="default"><a href="prvect_2.html#K6" title="PRVECT_2:func.6">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>1</sub></font>(<font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K5" title="PRVECT_2:func.5">.</a> <font color="Maroon" title="c5">i</font>) &amp; <span class="p1">(<span class="default"><a href="prvect_2.html#K7" title="PRVECT_2:func.7">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>  <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">comp</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K5" title="PRVECT_2:func.5">.</a> <font color="Maroon" title="c5">i</font></span>)</span> )
 <b>by </b><i><a class="txt" href="prvect_2.html#E2:23_1_1"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="prvect_2.html#D5" target="_self" title="PRVECT_2:def.5">Def5</a>, <a class="ref" href="prvect_2.html#D6" target="_self" title="PRVECT_2:def.6">Def6</a>, <a class="ref" href="prvect_1.html#T3" title="PRVECT_1:th.3">PRVECT_1:3</a></i>;<br/><b>hence </b><a NAME="E3:23_1_1_5_1"/>
( <span class="p1">(<span class="default"><a href="prvect_2.html#K7" title="PRVECT_2:func.7">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_2.html#K6" title="PRVECT_2:func.6">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_2.html#K6" title="PRVECT_2:func.6">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="setwiseo.html#NR1" title="SETWISEO:NR.1">has_a_unity</a>  )
 <b>by </b><i><a class="ref" href="prvect_1.html#T4" title="PRVECT_1:th.4">PRVECT_1:4</a>, <a class="ref" href="setwiseo.html#D2" title="SETWISEO:def.2">SETWISEO:def 2</a></i>;<br/></div><b>end;</b></div>
<a NAME="E3:23_1_1_5"/><b>then </b>
 <a href="prvect_1.html#K12" title="PRVECT_1:func.12">Frege</a> <span class="p1">(<span class="default"><a href="prvect_2.html#K7" title="PRVECT_2:func.7">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_2.html#K6" title="PRVECT_2:func.6">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>
 
<b>by </b><i><a class="ref" href="prvect_1.html#T28" title="PRVECT_1:th.28">PRVECT_1:28</a></i>;<br/>
<a NAME="E4:23_1_1_5"/><b>then </b>
<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_2.html#K6" title="PRVECT_2:func.6">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> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c3">x</font>,<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_2.html#K6" title="PRVECT_2:func.6">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>
 
<b>by </b><i><a class="ref" href="finseqop.html#D1" title="FINSEQOP:def.1">FINSEQOP:def 1</a></i>;<br/>
<b>hence </b><a NAME="E5:23_1_1_5"/>
<font color="Maroon" title="c3">x</font> <a href="rlvect_1.html#K1" title="RLVECT_1: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>
 <b>by </b><i><a class="txt" href="prvect_2.html#E8:23_1_1"><i><font color="Green" title="E21">A5</font></i></a>, <a class="ref" href="binop_1.html#D8" title="BINOP_1:def.8">BINOP_1:def 8</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E10:23_1_1"/>
(  <a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <font color="Maroon" title="c1">G</font> is <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">Abelian</a> &amp;  <a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <font color="Maroon" title="c1">G</font> is <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">add-associative</a> &amp;  <a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <font color="Maroon" title="c1">G</font> is <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">right_zeroed</a> &amp;  <a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <font color="Maroon" title="c1">G</font> is <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">right_complementable</a> &amp;  <a href="prvect_2.html#K10" title="PRVECT_2:func.10">product</a> <font color="Maroon" title="c1">G</font> is <a href="rlvect_1.html#V7" title="RLVECT_1:attr.7">RealLinearSpace-like</a> )
 <b>by </b><i><a class="txt" href="prvect_2.html#E4:23_1_1"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="prvect_2.html#E6:23_1_1"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="prvect_2.html#E8:23_1_1"><i><font color="Green" title="E21">A5</font></i></a>, <a class="txt" href="prvect_2.html#E15"><i><font color="Green" title="E13">Lm1</font></i></a>, <a class="txt" href="prvect_2.html#E16"><i><font color="Green" title="E14">Lm2</font></i></a>, <a class="txt" href="prvect_2.html#E19"><i><font color="Green" title="E17">Lm5</font></i></a></i>;<br/>


</div>
