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

<span class="kw">set </span><font color="Maroon" title="c2">X</font> =  <a href="bspace.html#K2" title="BSPACE:func.2">Z_2</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c3">X0</font> = <font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">Z0</font> =  <a href="nbvectsp.html#K2" title="NBVECTSP:func.2">ZeroB</a> <font color="Maroon" title="c1">n</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">ADD</font> =  <a href="nbvectsp.html#K1" title="NBVECTSP:func.1">XORB</a> <font color="Maroon" title="c1">n</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">LMLT</font> =  <a href="nbvectsp.html#K4" title="NBVECTSP:func.4">MLTB</a> <font color="Maroon" title="c1">n</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">XP</font> =  <a href="vectsp_1.html#G1" title="VECTSP_1:aggr.1">ModuleStr</a>(# <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K1" title="NBVECTSP:func.1">XORB</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K2" title="NBVECTSP:func.2">ZeroB</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K4" title="NBVECTSP:func.4">MLTB</a> <font color="Maroon" title="c1">n</font></span>)</span> #);<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">1X</font> =  <a href="margrel1.html#K7" title="MARGREL1:func.7">TRUE</a>  as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="bspace.html#K2" title="BSPACE:func.2">Z_2</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="bspace.html#T6" onmouseover="rs('bspace/T6')" onmouseout="rh()">BSPACE:6</a></span>;<br/>
<a NAME="E2:12_1_1"/><span class="lab"><font color="Green" title="E7">A1</font></span>: 
(  <a href="vectsp_1.html#G1" title="VECTSP_1:aggr.1">ModuleStr</a>(# <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K1" title="NBVECTSP:func.1">XORB</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K2" title="NBVECTSP:func.2">ZeroB</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K4" title="NBVECTSP:func.4">MLTB</a> <font color="Maroon" title="c1">n</font></span>)</span> #) is  <a href="vectsp_1.html#V9" title="VECTSP_1:attr.9">scalar-distributive</a>  &amp;  <a href="vectsp_1.html#G1" title="VECTSP_1:aggr.1">ModuleStr</a>(# <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K1" title="NBVECTSP:func.1">XORB</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K2" title="NBVECTSP:func.2">ZeroB</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K4" title="NBVECTSP:func.4">MLTB</a> <font color="Maroon" title="c1">n</font></span>)</span> #) is  <a href="vectsp_1.html#V8" title="VECTSP_1:attr.8">vector-distributive</a>  &amp;  <a href="vectsp_1.html#G1" title="VECTSP_1:aggr.1">ModuleStr</a>(# <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K1" title="NBVECTSP:func.1">XORB</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K2" title="NBVECTSP:func.2">ZeroB</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K4" title="NBVECTSP:func.4">MLTB</a> <font color="Maroon" title="c1">n</font></span>)</span> #) is  <a href="vectsp_1.html#V10" title="VECTSP_1:attr.10">scalar-associative</a>  &amp;  <a href="vectsp_1.html#G1" title="VECTSP_1:aggr.1">ModuleStr</a>(# <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K1" title="NBVECTSP:func.1">XORB</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K2" title="NBVECTSP:func.2">ZeroB</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K4" title="NBVECTSP:func.4">MLTB</a> <font color="Maroon" title="c1">n</font></span>)</span> #) is  <a href="vectsp_1.html#V11" title="VECTSP_1:attr.11">scalar-unital</a>  )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="12_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">thus </span><a NAME="E1:12_1_1_1"/>
 <a href="vectsp_1.html#G1" title="VECTSP_1:aggr.1">ModuleStr</a>(# <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K1" title="NBVECTSP:func.1">XORB</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K2" title="NBVECTSP:func.2">ZeroB</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K4" title="NBVECTSP:func.4">MLTB</a> <font color="Maroon" title="c1">n</font></span>)</span> #) is  <a href="vectsp_1.html#V9" title="VECTSP_1:attr.9">scalar-distributive</a> 
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="vectsp_1.html#G1" title="VECTSP_1:aggr.1">ModuleStr</a>(# <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K1" title="NBVECTSP:func.1">XORB</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K2" title="NBVECTSP:func.2">ZeroB</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K4" title="NBVECTSP:func.4">MLTB</a> <font color="Maroon" title="c1">n</font></span>)</span> #) is  <a href="vectsp_1.html#V8" title="VECTSP_1:attr.8">vector-distributive</a>  &amp;  <a href="vectsp_1.html#G1" title="VECTSP_1:aggr.1">ModuleStr</a>(# <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K1" title="NBVECTSP:func.1">XORB</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K2" title="NBVECTSP:func.2">ZeroB</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K4" title="NBVECTSP:func.4">MLTB</a> <font color="Maroon" title="c1">n</font></span>)</span> #) is  <a href="vectsp_1.html#V10" title="VECTSP_1:attr.10">scalar-associative</a>  &amp;  <a href="vectsp_1.html#G1" title="VECTSP_1:aggr.1">ModuleStr</a>(# <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K1" title="NBVECTSP:func.1">XORB</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K2" title="NBVECTSP:func.2">ZeroB</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K4" title="NBVECTSP:func.4">MLTB</a> <font color="Maroon" title="c1">n</font></span>)</span> #) is  <a href="vectsp_1.html#V11" title="VECTSP_1:attr.11">scalar-unital</a>  )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="12_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">x</font>, <font color="Maroon" title="c10">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="bspace.html#K2" title="BSPACE:func.2">Z_2</a>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="vectsp_1.html#D14" onmouseover="rs('vectsp_1/D14')" onmouseout="rh()">VECTSP_1:def 14</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive">b<sub>1</sub></font> being    <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 <a href="vectsp_1.html#G1" title="VECTSP_1:aggr.1">ModuleStr</a>(# <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K1" title="NBVECTSP:func.1">XORB</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K2" title="NBVECTSP:func.2">ZeroB</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K4" title="NBVECTSP:func.4">MLTB</a> <font color="Maroon" title="c1">n</font></span>)</span> #) holds  <span class="p1">(<span class="default"><font color="Maroon" title="c9">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c10">y</font></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">x</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">y</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c11">v</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="vectsp_1.html#G1" title="VECTSP_1:aggr.1">ModuleStr</a>(# <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K1" title="NBVECTSP:func.1">XORB</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K2" title="NBVECTSP:func.2">ZeroB</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K4" title="NBVECTSP:func.4">MLTB</a> <font color="Maroon" title="c1">n</font></span>)</span> #); <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c9">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c10">y</font></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c11">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">x</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c11">v</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">y</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c11">v</font></span>)</span></span><br/>



<span class="kw">reconsider </span><font color="Maroon" title="c12">v1</font> = <font color="Maroon" title="c11">v</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">xyv1</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c9">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c10">y</font></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c11">v</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">xv</font> = <font color="Maroon" title="c9">x</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c11">v</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">yv</font> = <font color="Maroon" title="c10">y</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c11">v</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c16">xyv2</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c9">x</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c11">v</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">y</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c11">v</font></span>)</span> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a> ;<br/>
<a NAME="E6:12_1_1_1_1"/><span class="lab"><font color="Green" title="E8">A2</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">xyv1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D7" onmouseover="rs('card_1/D7')" onmouseout="rh()">CARD_1:def 7</a></span>;<br/>
<a NAME="E7:12_1_1_1_1"/><span class="lab"><font color="Green" title="E9">A3</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">xyv2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D7" onmouseover="rs('card_1/D7')" onmouseout="rh()">CARD_1:def 7</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="12_1_1_1_1_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">xyv1</font> holds <br/><font color="Maroon" title="c13">xyv1</font> <a href="descip_1.html#K10" title="DESCIP_1:func.10">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">xyv2</font> <a href="descip_1.html#K10" title="DESCIP_1:func.10">.</a> <font color="Olive" title="b1">i</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c17">i</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">i</font> &amp; <font color="Maroon" title="c17">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">xyv1</font> implies <font color="Maroon" title="c13">xyv1</font> <a href="descip_1.html#K10" title="DESCIP_1:func.10">.</a> <font color="Maroon" title="c17">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">xyv2</font> <a href="descip_1.html#K10" title="DESCIP_1:func.10">.</a> <font color="Maroon" title="c17">i</font> )</span><br/><span class="kw">assume </span><a NAME="E1:12_1_1_1_1_1"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">i</font> &amp; <font color="Maroon" title="c17">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">xyv1</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">xyv1</font> <a href="descip_1.html#K10" title="DESCIP_1:func.10">.</a> <font color="Maroon" title="c17">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">xyv2</font> <a href="descip_1.html#K10" title="DESCIP_1:func.10">.</a> <font color="Maroon" title="c17">i</font></span><br/><a NAME="E2:12_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E10">A4</font></span>: 
<font color="Maroon" title="c17">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:12_1_1_1_1"><span class="lab"><font color="Green" title="E8">A2</font></span></a></span>;<br/><a NAME="E3:12_1_1_1_1_1"/><span class="lab"><font color="Green" title="E11">A5</font></span>: 
<font color="Maroon" title="c14">xv</font> <a href="descip_1.html#K10" title="DESCIP_1:func.10">.</a> <font color="Maroon" title="c17">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x</font> <a href="xboolean.html#K4" title="XBOOLEAN:func.4">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">v1</font> <a href="descip_1.html#K10" title="DESCIP_1:func.10">.</a> <font color="Maroon" title="c17">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nbvectsp.html#D4" target="_self" onmouseover="rs('nbvectsp/D4')" onmouseout="rh()">Def4</a>, <a class="txt" href="#E2:12_1_1_1_1_1"><span class="lab"><font color="Green" title="E10">A4</font></span></a>, <a class="ref" href="bspace.html#T3" onmouseover="rs('bspace/T3')" onmouseout="rh()">BSPACE:3</a></span>;<br/><a NAME="E4:12_1_1_1_1_1"/><span class="lab"><font color="Green" title="E12">A6</font></span>: 
<font color="Maroon" title="c15">yv</font> <a href="descip_1.html#K10" title="DESCIP_1:func.10">.</a> <font color="Maroon" title="c17">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y</font> <a href="xboolean.html#K4" title="XBOOLEAN:func.4">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">v1</font> <a href="descip_1.html#K10" title="DESCIP_1:func.10">.</a> <font color="Maroon" title="c17">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nbvectsp.html#D4" target="_self" onmouseover="rs('nbvectsp/D4')" onmouseout="rh()">Def4</a>, <a class="txt" href="#E2:12_1_1_1_1_1"><span class="lab"><font color="Green" title="E10">A4</font></span></a>, <a class="ref" href="bspace.html#T3" onmouseover="rs('bspace/T3')" onmouseout="rh()">BSPACE:3</a></span>;<br/><span class="kw">thus </span><font color="Maroon" title="c13">xyv1</font> <a href="descip_1.html#K10" title="DESCIP_1:func.10">.</a> <font color="Maroon" title="c17">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">x</font> <a href="xboolean.html#K10" title="XBOOLEAN:func.10">'xor'</a> <font color="Maroon" title="c10">y</font></span>)</span> <a href="xboolean.html#K4" title="XBOOLEAN:func.4">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">v1</font> <a href="descip_1.html#K10" title="DESCIP_1:func.10">.</a> <font color="Maroon" title="c17">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nbvectsp.html#D4" target="_self" onmouseover="rs('nbvectsp/D4')" onmouseout="rh()">Def4</a>, <a class="txt" href="#E2:12_1_1_1_1_1"><span class="lab"><font color="Green" title="E10">A4</font></span></a>, <a class="ref" href="bspace.html#T3" onmouseover="rs('bspace/T3')" onmouseout="rh()">BSPACE:3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">xv</font> <a href="descip_1.html#K10" title="DESCIP_1:func.10">.</a> <font color="Maroon" title="c17">i</font></span>)</span> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">yv</font> <a href="descip_1.html#K10" title="DESCIP_1:func.10">.</a> <font color="Maroon" title="c17">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboolean.html#T75" onmouseover="rs('xboolean/T75')" onmouseout="rh()">XBOOLEAN:75</a>, <a class="txt" href="#E3:12_1_1_1_1_1"><span class="lab"><font color="Green" title="E11">A5</font></span></a>, <a class="txt" href="#E4:12_1_1_1_1_1"><span class="lab"><font color="Green" title="E12">A6</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="descip_1.html#K11" title="DESCIP_1:func.11">Op-XOR</a> (<font color="Maroon" title="c14">xv</font>,<font color="Maroon" title="c15">yv</font>)</span>)</span> <a href="descip_1.html#K10" title="DESCIP_1:func.10">.</a> <font color="Maroon" title="c17">i</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="descip_1.html#D4" onmouseover="rs('descip_1/D4')" onmouseout="rh()">DESCIP_1:def 4</a>, <a class="txt" href="#E2:12_1_1_1_1_1"><span class="lab"><font color="Green" title="E10">A4</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c16">xyv2</font> <a href="descip_1.html#K10" title="DESCIP_1:func.10">.</a> <font color="Maroon" title="c17">i</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nbvectsp.html#D1" target="_self" onmouseover="rs('nbvectsp/D1')" onmouseout="rh()">Def1</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E9:12_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c10">y</font></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c11">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">x</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c11">v</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">y</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c11">v</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T14" onmouseover="rs('finseq_1/T14')" onmouseout="rh()">FINSEQ_1:14</a>, <a class="ref" href="card_1.html#D7" onmouseover="rs('card_1/D7')" onmouseout="rh()">CARD_1:def 7</a>, <a class="txt" href="#E7:12_1_1_1_1"><span class="lab"><font color="Green" title="E9">A3</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E2:12_1_1_1"/>
 <a href="vectsp_1.html#G1" title="VECTSP_1:aggr.1">ModuleStr</a>(# <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K1" title="NBVECTSP:func.1">XORB</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K2" title="NBVECTSP:func.2">ZeroB</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K4" title="NBVECTSP:func.4">MLTB</a> <font color="Maroon" title="c1">n</font></span>)</span> #) is  <a href="vectsp_1.html#V8" title="VECTSP_1:attr.8">vector-distributive</a> 
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="vectsp_1.html#G1" title="VECTSP_1:aggr.1">ModuleStr</a>(# <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K1" title="NBVECTSP:func.1">XORB</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K2" title="NBVECTSP:func.2">ZeroB</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K4" title="NBVECTSP:func.4">MLTB</a> <font color="Maroon" title="c1">n</font></span>)</span> #) is  <a href="vectsp_1.html#V10" title="VECTSP_1:attr.10">scalar-associative</a>  &amp;  <a href="vectsp_1.html#G1" title="VECTSP_1:aggr.1">ModuleStr</a>(# <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K1" title="NBVECTSP:func.1">XORB</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K2" title="NBVECTSP:func.2">ZeroB</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K4" title="NBVECTSP:func.4">MLTB</a> <font color="Maroon" title="c1">n</font></span>)</span> #) is  <a href="vectsp_1.html#V11" title="VECTSP_1:attr.11">scalar-unital</a>  )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="12_1_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="bspace.html#K2" title="BSPACE:func.2">Z_2</a>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="vectsp_1.html#D13" onmouseover="rs('vectsp_1/D13')" onmouseout="rh()">VECTSP_1:def 13</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being    <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 <a href="vectsp_1.html#G1" title="VECTSP_1:aggr.1">ModuleStr</a>(# <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K1" title="NBVECTSP:func.1">XORB</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K2" title="NBVECTSP:func.2">ZeroB</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K4" title="NBVECTSP:func.4">MLTB</a> <font color="Maroon" title="c1">n</font></span>)</span> #) holds  <font color="Maroon" title="c9">x</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">x</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">x</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c10">v</font>, <font color="Maroon" title="c11">w</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="vectsp_1.html#G1" title="VECTSP_1:aggr.1">ModuleStr</a>(# <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K1" title="NBVECTSP:func.1">XORB</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K2" title="NBVECTSP:func.2">ZeroB</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K4" title="NBVECTSP:func.4">MLTB</a> <font color="Maroon" title="c1">n</font></span>)</span> #); <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">x</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c11">w</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">x</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c10">v</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">x</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c11">w</font></span>)</span></span><br/>



<span class="kw">reconsider </span><font color="Maroon" title="c12">v1</font> = <font color="Maroon" title="c10">v</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">w1</font> = <font color="Maroon" title="c11">w</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">xvw1</font> = <font color="Maroon" title="c9">x</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c11">w</font></span>)</span> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">vw</font> = <font color="Maroon" title="c10">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c11">w</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c16">xv</font> = <font color="Maroon" title="c9">x</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c10">v</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c17">xw</font> = <font color="Maroon" title="c9">x</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c11">w</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c18">xvw2</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c9">x</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c10">v</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">x</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c11">w</font></span>)</span> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a> ;<br/>
<a NAME="E8:12_1_1_1_2"/><span class="lab"><font color="Green" title="E8">A7</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">xvw1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D7" onmouseover="rs('card_1/D7')" onmouseout="rh()">CARD_1:def 7</a></span>;<br/>
<a NAME="E9:12_1_1_1_2"/><span class="lab"><font color="Green" title="E9">A8</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">xvw2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D7" onmouseover="rs('card_1/D7')" onmouseout="rh()">CARD_1:def 7</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="12_1_1_1_2_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">xvw1</font> holds <br/><font color="Maroon" title="c18">xvw2</font> <a href="descip_1.html#K10" title="DESCIP_1:func.10">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">xvw1</font> <a href="descip_1.html#K10" title="DESCIP_1:func.10">.</a> <font color="Olive" title="b1">i</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c19">i</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">i</font> &amp; <font color="Maroon" title="c19">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">xvw1</font> implies <font color="Maroon" title="c18">xvw2</font> <a href="descip_1.html#K10" title="DESCIP_1:func.10">.</a> <font color="Maroon" title="c19">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">xvw1</font> <a href="descip_1.html#K10" title="DESCIP_1:func.10">.</a> <font color="Maroon" title="c19">i</font> )</span><br/><span class="kw">assume </span><a NAME="E1:12_1_1_1_2_1"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">i</font> &amp; <font color="Maroon" title="c19">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">xvw1</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c18">xvw2</font> <a href="descip_1.html#K10" title="DESCIP_1:func.10">.</a> <font color="Maroon" title="c19">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">xvw1</font> <a href="descip_1.html#K10" title="DESCIP_1:func.10">.</a> <font color="Maroon" title="c19">i</font></span><br/><a NAME="E2:12_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E10">A9</font></span>: 
<font color="Maroon" title="c19">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:12_1_1_1_2"><span class="lab"><font color="Green" title="E8">A7</font></span></a></span>;<br/><a NAME="E3:12_1_1_1_2_1"/><span class="lab"><font color="Green" title="E11">A10</font></span>: 
<font color="Maroon" title="c16">xv</font> <a href="descip_1.html#K10" title="DESCIP_1:func.10">.</a> <font color="Maroon" title="c19">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x</font> <a href="xboolean.html#K4" title="XBOOLEAN:func.4">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">v1</font> <a href="descip_1.html#K10" title="DESCIP_1:func.10">.</a> <font color="Maroon" title="c19">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nbvectsp.html#D4" target="_self" onmouseover="rs('nbvectsp/D4')" onmouseout="rh()">Def4</a>, <a class="txt" href="#E2:12_1_1_1_2_1"><span class="lab"><font color="Green" title="E10">A9</font></span></a>, <a class="ref" href="bspace.html#T3" onmouseover="rs('bspace/T3')" onmouseout="rh()">BSPACE:3</a></span>;<br/><a NAME="E4:12_1_1_1_2_1"/><span class="lab"><font color="Green" title="E12">A11</font></span>: 
<font color="Maroon" title="c17">xw</font> <a href="descip_1.html#K10" title="DESCIP_1:func.10">.</a> <font color="Maroon" title="c19">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x</font> <a href="xboolean.html#K4" title="XBOOLEAN:func.4">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">w1</font> <a href="descip_1.html#K10" title="DESCIP_1:func.10">.</a> <font color="Maroon" title="c19">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nbvectsp.html#D4" target="_self" onmouseover="rs('nbvectsp/D4')" onmouseout="rh()">Def4</a>, <a class="txt" href="#E2:12_1_1_1_2_1"><span class="lab"><font color="Green" title="E10">A9</font></span></a>, <a class="ref" href="bspace.html#T3" onmouseover="rs('bspace/T3')" onmouseout="rh()">BSPACE:3</a></span>;<br/><span class="lab"><font color="Green" title="E13">A12</font></span>: <font color="Maroon" title="c15">vw</font> <a href="descip_1.html#K10" title="DESCIP_1:func.10">.</a> <font color="Maroon" title="c19">i</font> = 
<span class="p1">(<span class="default"><a href="descip_1.html#K11" title="DESCIP_1:func.11">Op-XOR</a> (<font color="Maroon" title="c12">v1</font>,<font color="Maroon" title="c13">w1</font>)</span>)</span> <a href="descip_1.html#K10" title="DESCIP_1:func.10">.</a> <font color="Maroon" title="c19">i</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nbvectsp.html#D1" target="_self" onmouseover="rs('nbvectsp/D1')" onmouseout="rh()">Def1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">v1</font> <a href="descip_1.html#K10" title="DESCIP_1:func.10">.</a> <font color="Maroon" title="c19">i</font></span>)</span> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">w1</font> <a href="descip_1.html#K10" title="DESCIP_1:func.10">.</a> <font color="Maroon" title="c19">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="descip_1.html#D4" onmouseover="rs('descip_1/D4')" onmouseout="rh()">DESCIP_1:def 4</a>, <a class="txt" href="#E2:12_1_1_1_2_1"><span class="lab"><font color="Green" title="E10">A9</font></span></a></span>
;<br/><span class="kw">thus </span><font color="Maroon" title="c18">xvw2</font> <a href="descip_1.html#K10" title="DESCIP_1:func.10">.</a> <font color="Maroon" title="c19">i</font> = 
<span class="p1">(<span class="default"><a href="descip_1.html#K11" title="DESCIP_1:func.11">Op-XOR</a> (<font color="Maroon" title="c16">xv</font>,<font color="Maroon" title="c17">xw</font>)</span>)</span> <a href="descip_1.html#K10" title="DESCIP_1:func.10">.</a> <font color="Maroon" title="c19">i</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nbvectsp.html#D1" target="_self" onmouseover="rs('nbvectsp/D1')" onmouseout="rh()">Def1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">x</font> <a href="xboolean.html#K4" title="XBOOLEAN:func.4">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">v1</font> <a href="descip_1.html#K10" title="DESCIP_1:func.10">.</a> <font color="Maroon" title="c19">i</font></span>)</span></span>)</span> <a href="xboolean.html#K10" title="XBOOLEAN:func.10">'xor'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">x</font> <a href="xboolean.html#K4" title="XBOOLEAN:func.4">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">w1</font> <a href="descip_1.html#K10" title="DESCIP_1:func.10">.</a> <font color="Maroon" title="c19">i</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="descip_1.html#D4" onmouseover="rs('descip_1/D4')" onmouseout="rh()">DESCIP_1:def 4</a>, <a class="txt" href="#E2:12_1_1_1_2_1"><span class="lab"><font color="Green" title="E10">A9</font></span></a>, <a class="txt" href="#E3:12_1_1_1_2_1"><span class="lab"><font color="Green" title="E11">A10</font></span></a>, <a class="txt" href="#E4:12_1_1_1_2_1"><span class="lab"><font color="Green" title="E12">A11</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c9">x</font> <a href="xboolean.html#K4" title="XBOOLEAN:func.4">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">vw</font> <a href="descip_1.html#K10" title="DESCIP_1:func.10">.</a> <font color="Maroon" title="c19">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboolean.html#T75" onmouseover="rs('xboolean/T75')" onmouseout="rh()">XBOOLEAN:75</a>, <a class="txt" href="#E5:12_1_1_1_2_1"><span class="lab"><font color="Green" title="E13">A12</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c14">xvw1</font> <a href="descip_1.html#K10" title="DESCIP_1:func.10">.</a> <font color="Maroon" title="c19">i</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nbvectsp.html#D4" target="_self" onmouseover="rs('nbvectsp/D4')" onmouseout="rh()">Def4</a>, <a class="txt" href="#E2:12_1_1_1_2_1"><span class="lab"><font color="Green" title="E10">A9</font></span></a>, <a class="ref" href="bspace.html#T3" onmouseover="rs('bspace/T3')" onmouseout="rh()">BSPACE:3</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E11:12_1_1_1_2"/>
<font color="Maroon" title="c9">x</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c11">w</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">x</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c10">v</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">x</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c11">w</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T14" onmouseover="rs('finseq_1/T14')" onmouseout="rh()">FINSEQ_1:14</a>, <a class="ref" href="card_1.html#D7" onmouseover="rs('card_1/D7')" onmouseout="rh()">CARD_1:def 7</a>, <a class="txt" href="#E9:12_1_1_1_2"><span class="lab"><font color="Green" title="E9">A8</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E3:12_1_1_1"/>
 <a href="vectsp_1.html#G1" title="VECTSP_1:aggr.1">ModuleStr</a>(# <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K1" title="NBVECTSP:func.1">XORB</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K2" title="NBVECTSP:func.2">ZeroB</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K4" title="NBVECTSP:func.4">MLTB</a> <font color="Maroon" title="c1">n</font></span>)</span> #) is  <a href="vectsp_1.html#V10" title="VECTSP_1:attr.10">scalar-associative</a> 
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="vectsp_1.html#G1" title="VECTSP_1:aggr.1">ModuleStr</a>(# <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K1" title="NBVECTSP:func.1">XORB</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K2" title="NBVECTSP:func.2">ZeroB</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K4" title="NBVECTSP:func.4">MLTB</a> <font color="Maroon" title="c1">n</font></span>)</span> #) is  <a href="vectsp_1.html#V11" title="VECTSP_1:attr.11">scalar-unital</a> </span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="12_1_1_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">x</font>, <font color="Maroon" title="c10">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="bspace.html#K2" title="BSPACE:func.2">Z_2</a>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="vectsp_1.html#D15" onmouseover="rs('vectsp_1/D15')" onmouseout="rh()">VECTSP_1:def 15</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive">b<sub>1</sub></font> being    <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 <a href="vectsp_1.html#G1" title="VECTSP_1:aggr.1">ModuleStr</a>(# <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K1" title="NBVECTSP:func.1">XORB</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K2" title="NBVECTSP:func.2">ZeroB</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K4" title="NBVECTSP:func.4">MLTB</a> <font color="Maroon" title="c1">n</font></span>)</span> #) holds  <span class="p1">(<span class="default"><font color="Maroon" title="c9">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c10">y</font></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">y</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c11">v</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="vectsp_1.html#G1" title="VECTSP_1:aggr.1">ModuleStr</a>(# <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K1" title="NBVECTSP:func.1">XORB</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K2" title="NBVECTSP:func.2">ZeroB</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K4" title="NBVECTSP:func.4">MLTB</a> <font color="Maroon" title="c1">n</font></span>)</span> #); <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c9">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c10">y</font></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c11">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">y</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c11">v</font></span>)</span></span><br/>



<span class="kw">reconsider </span><font color="Maroon" title="c12">v1</font> = <font color="Maroon" title="c11">v</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">xyv1</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c9">x</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c10">y</font></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c11">v</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">yv1</font> = <font color="Maroon" title="c10">y</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c11">v</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">xyv2</font> = <font color="Maroon" title="c9">x</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">y</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c11">v</font></span>)</span> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a> ;<br/>
<a NAME="E5:12_1_1_1_3"/><span class="lab"><font color="Green" title="E8">A13</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">xyv1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D7" onmouseover="rs('card_1/D7')" onmouseout="rh()">CARD_1:def 7</a></span>;<br/>
<a NAME="E6:12_1_1_1_3"/><span class="lab"><font color="Green" title="E9">A14</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">xyv2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D7" onmouseover="rs('card_1/D7')" onmouseout="rh()">CARD_1:def 7</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="12_1_1_1_3_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">xyv1</font> holds <br/><font color="Maroon" title="c15">xyv2</font> <a href="descip_1.html#K10" title="DESCIP_1:func.10">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">xyv1</font> <a href="descip_1.html#K10" title="DESCIP_1:func.10">.</a> <font color="Olive" title="b1">i</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c16">i</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">i</font> &amp; <font color="Maroon" title="c16">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">xyv1</font> implies <font color="Maroon" title="c15">xyv2</font> <a href="descip_1.html#K10" title="DESCIP_1:func.10">.</a> <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">xyv1</font> <a href="descip_1.html#K10" title="DESCIP_1:func.10">.</a> <font color="Maroon" title="c16">i</font> )</span><br/><span class="kw">assume </span><a NAME="E1:12_1_1_1_3_1"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">i</font> &amp; <font color="Maroon" title="c16">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">xyv1</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c15">xyv2</font> <a href="descip_1.html#K10" title="DESCIP_1:func.10">.</a> <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">xyv1</font> <a href="descip_1.html#K10" title="DESCIP_1:func.10">.</a> <font color="Maroon" title="c16">i</font></span><br/><a NAME="E2:12_1_1_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E10">A15</font></span>: 
<font color="Maroon" title="c16">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:12_1_1_1_3"><span class="lab"><font color="Green" title="E8">A13</font></span></a></span>;<br/><a NAME="E3:12_1_1_1_3_1"/><span class="lab"><font color="Green" title="E11">A16</font></span>: 
<font color="Maroon" title="c14">yv1</font> <a href="descip_1.html#K10" title="DESCIP_1:func.10">.</a> <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y</font> <a href="xboolean.html#K4" title="XBOOLEAN:func.4">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">v1</font> <a href="descip_1.html#K10" title="DESCIP_1:func.10">.</a> <font color="Maroon" title="c16">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nbvectsp.html#D4" target="_self" onmouseover="rs('nbvectsp/D4')" onmouseout="rh()">Def4</a>, <a class="txt" href="#E2:12_1_1_1_3_1"><span class="lab"><font color="Green" title="E10">A15</font></span></a>, <a class="ref" href="bspace.html#T3" onmouseover="rs('bspace/T3')" onmouseout="rh()">BSPACE:3</a></span>;<br/><span class="kw">thus </span><font color="Maroon" title="c15">xyv2</font> <a href="descip_1.html#K10" title="DESCIP_1:func.10">.</a> <font color="Maroon" title="c16">i</font> = 
<font color="Maroon" title="c9">x</font> <a href="xboolean.html#K4" title="XBOOLEAN:func.4">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">yv1</font> <a href="descip_1.html#K10" title="DESCIP_1:func.10">.</a> <font color="Maroon" title="c16">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nbvectsp.html#D4" target="_self" onmouseover="rs('nbvectsp/D4')" onmouseout="rh()">Def4</a>, <a class="txt" href="#E2:12_1_1_1_3_1"><span class="lab"><font color="Green" title="E10">A15</font></span></a>, <a class="ref" href="bspace.html#T3" onmouseover="rs('bspace/T3')" onmouseout="rh()">BSPACE:3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">x</font> <a href="xboolean.html#K4" title="XBOOLEAN:func.4">'&amp;'</a> <font color="Maroon" title="c10">y</font></span>)</span> <a href="xboolean.html#K4" title="XBOOLEAN:func.4">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">v1</font> <a href="descip_1.html#K10" title="DESCIP_1:func.10">.</a> <font color="Maroon" title="c16">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:12_1_1_1_3_1"><span class="lab"><font color="Green" title="E11">A16</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c13">xyv1</font> <a href="descip_1.html#K10" title="DESCIP_1:func.10">.</a> <font color="Maroon" title="c16">i</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nbvectsp.html#D4" target="_self" onmouseover="rs('nbvectsp/D4')" onmouseout="rh()">Def4</a>, <a class="txt" href="#E2:12_1_1_1_3_1"><span class="lab"><font color="Green" title="E10">A15</font></span></a>, <a class="ref" href="bspace.html#T3" onmouseover="rs('bspace/T3')" onmouseout="rh()">BSPACE:3</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E8:12_1_1_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c10">y</font></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c11">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">y</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c11">v</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T14" onmouseover="rs('finseq_1/T14')" onmouseout="rh()">FINSEQ_1:14</a>, <a class="ref" href="card_1.html#D7" onmouseover="rs('card_1/D7')" onmouseout="rh()">CARD_1:def 7</a>, <a class="txt" href="#E6:12_1_1_1_3"><span class="lab"><font color="Green" title="E9">A14</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E4:12_1_1_1"/>
 <a href="vectsp_1.html#G1" title="VECTSP_1:aggr.1">ModuleStr</a>(# <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K1" title="NBVECTSP:func.1">XORB</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K2" title="NBVECTSP:func.2">ZeroB</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K4" title="NBVECTSP:func.4">MLTB</a> <font color="Maroon" title="c1">n</font></span>)</span> #) is  <a href="vectsp_1.html#V11" title="VECTSP_1:attr.11">scalar-unital</a> 
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="12_1_1_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">v</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="vectsp_1.html#G1" title="VECTSP_1:aggr.1">ModuleStr</a>(# <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K1" title="NBVECTSP:func.1">XORB</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K2" title="NBVECTSP:func.2">ZeroB</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K4" title="NBVECTSP:func.4">MLTB</a> <font color="Maroon" title="c1">n</font></span>)</span> #); <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="vectsp_1.html#D16" onmouseover="rs('vectsp_1/D16')" onmouseout="rh()">VECTSP_1:def 16</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <a href="bspace.html#K2" title="BSPACE:func.2">Z_2</a></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c9">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">v</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c10">v1</font> = <font color="Maroon" title="c9">v</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">w1</font> = <span class="p1">(<span class="default"><a href="nbvectsp.html#K4" title="NBVECTSP:func.4">MLTB</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c8">1X</font>,<font color="Maroon" title="c10">v1</font>) as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a> ;<br/>
<a NAME="E3:12_1_1_1_4"/><span class="lab"><font color="Green" title="E8">A17</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D7" onmouseover="rs('card_1/D7')" onmouseout="rh()">CARD_1:def 7</a></span>;<br/>
<a NAME="E4:12_1_1_1_4"/><span class="lab"><font color="Green" title="E9">A18</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D7" onmouseover="rs('card_1/D7')" onmouseout="rh()">CARD_1:def 7</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="12_1_1_1_4_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">v1</font> holds <br/><font color="Maroon" title="c11">w1</font> <a href="descip_1.html#K10" title="DESCIP_1:func.10">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">v1</font> <a href="descip_1.html#K10" title="DESCIP_1:func.10">.</a> <font color="Olive" title="b1">i</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c12">i</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">i</font> &amp; <font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">v1</font> implies <font color="Maroon" title="c11">w1</font> <a href="descip_1.html#K10" title="DESCIP_1:func.10">.</a> <font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">v1</font> <a href="descip_1.html#K10" title="DESCIP_1:func.10">.</a> <font color="Maroon" title="c12">i</font> )</span><br/><span class="kw">assume </span><a NAME="E1:12_1_1_1_4_1"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">i</font> &amp; <font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">v1</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">w1</font> <a href="descip_1.html#K10" title="DESCIP_1:func.10">.</a> <font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">v1</font> <a href="descip_1.html#K10" title="DESCIP_1:func.10">.</a> <font color="Maroon" title="c12">i</font></span><br/><a NAME="E2:12_1_1_1_4_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:12_1_1_1_4"><span class="lab"><font color="Green" title="E8">A17</font></span></a></span>;<br/><span class="kw">hence </span><font color="Maroon" title="c11">w1</font> <a href="descip_1.html#K10" title="DESCIP_1:func.10">.</a> <font color="Maroon" title="c12">i</font> = 
<a href="margrel1.html#K7" title="MARGREL1:func.7">TRUE</a> <a href="margrel1.html#K9" title="MARGREL1:func.9">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">v1</font> <a href="descip_1.html#K10" title="DESCIP_1:func.10">.</a> <font color="Maroon" title="c12">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nbvectsp.html#D4" target="_self" onmouseover="rs('nbvectsp/D4')" onmouseout="rh()">Def4</a></span>
<br/>.= 
<font color="Maroon" title="c10">v1</font> <a href="descip_1.html#K10" title="DESCIP_1:func.10">.</a> <font color="Maroon" title="c12">i</font>

;<br/> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E6:12_1_1_1_4"/>
<span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <a href="bspace.html#K2" title="BSPACE:func.2">Z_2</a></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c9">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">v</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T14" onmouseover="rs('finseq_1/T14')" onmouseout="rh()">FINSEQ_1:14</a>, <a class="ref" href="card_1.html#D7" onmouseover="rs('card_1/D7')" onmouseout="rh()">CARD_1:def 7</a>, <a class="txt" href="#E4:12_1_1_1_4"><span class="lab"><font color="Green" title="E9">A18</font></span></a>, <a class="ref" href="bspace.html#T6" onmouseover="rs('bspace/T6')" onmouseout="rh()">BSPACE:6</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>


</div><span class="kw">end;</span></div>
<a NAME="E3:12_1_1"/><span class="lab"><font color="Green" title="E8">A19</font></span>: 
 <a href="vectsp_1.html#G1" title="VECTSP_1:aggr.1">ModuleStr</a>(# <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K1" title="NBVECTSP:func.1">XORB</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K2" title="NBVECTSP:func.2">ZeroB</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K4" title="NBVECTSP:func.4">MLTB</a> <font color="Maroon" title="c1">n</font></span>)</span> #) is  <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="12_1_1_2"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:12_1_1_2"/>
 for <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">v</font>, <font color="Olive" title="b3">w</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="vectsp_1.html#G1" title="VECTSP_1:aggr.1">ModuleStr</a>(# <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K1" title="NBVECTSP:func.1">XORB</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K2" title="NBVECTSP:func.2">ZeroB</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K4" title="NBVECTSP:func.4">MLTB</a> <font color="Maroon" title="c1">n</font></span>)</span> #) holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">u</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b2">v</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b3">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">u</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b3">w</font></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="12_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">u</font>, <font color="Maroon" title="c10">v</font>, <font color="Maroon" title="c11">w</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="vectsp_1.html#G1" title="VECTSP_1:aggr.1">ModuleStr</a>(# <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K1" title="NBVECTSP:func.1">XORB</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K2" title="NBVECTSP:func.2">ZeroB</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K4" title="NBVECTSP:func.4">MLTB</a> <font color="Maroon" title="c1">n</font></span>)</span> #); <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c9">u</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c10">v</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c11">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">u</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c11">w</font></span>)</span></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c12">u1</font> = <font color="Maroon" title="c9">u</font>, <font color="Maroon" title="c13">v1</font> = <font color="Maroon" title="c10">v</font>, <font color="Maroon" title="c14">w1</font> = <font color="Maroon" title="c11">w</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a> ;<br/>
<a NAME="E2:12_1_1_2_1"/><span class="lab"><font color="Green" title="E9">A20</font></span>: 
<font color="Maroon" title="c9">u</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c10">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="descip_1.html#K11" title="DESCIP_1:func.11">Op-XOR</a> (<font color="Maroon" title="c12">u1</font>,<font color="Maroon" title="c13">v1</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nbvectsp.html#D1" target="_self" onmouseover="rs('nbvectsp/D1')" onmouseout="rh()">Def1</a></span>;<br/>
<a NAME="E3:12_1_1_2_1"/><span class="lab"><font color="Green" title="E10">A21</font></span>: 
<font color="Maroon" title="c10">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c11">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="descip_1.html#K11" title="DESCIP_1:func.11">Op-XOR</a> (<font color="Maroon" title="c13">v1</font>,<font color="Maroon" title="c14">w1</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nbvectsp.html#D1" target="_self" onmouseover="rs('nbvectsp/D1')" onmouseout="rh()">Def1</a></span>;<br/>
<span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c9">u</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c10">v</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c11">w</font> = 
 <a href="descip_1.html#K11" title="DESCIP_1:func.11">Op-XOR</a> (<span class="p1">(<span class="default"><a href="descip_1.html#K11" title="DESCIP_1:func.11">Op-XOR</a> (<font color="Maroon" title="c12">u1</font>,<font color="Maroon" title="c13">v1</font>)</span>)</span>,<font color="Maroon" title="c14">w1</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:12_1_1_2_1"><span class="lab"><font color="Green" title="E9">A20</font></span></a>, <a class="ref" href="nbvectsp.html#D1" target="_self" onmouseover="rs('nbvectsp/D1')" onmouseout="rh()">Def1</a></span>
<br/>.= 
 <a href="descip_1.html#K11" title="DESCIP_1:func.11">Op-XOR</a> (<font color="Maroon" title="c12">u1</font>,<span class="p1">(<span class="default"><a href="descip_1.html#K11" title="DESCIP_1:func.11">Op-XOR</a> (<font color="Maroon" title="c13">v1</font>,<font color="Maroon" title="c14">w1</font>)</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="nbvectsp.html#T1" target="_self" onmouseover="rs('nbvectsp/T1')" onmouseout="rh()">Th1</a></span>
<br/>.= 
<font color="Maroon" title="c9">u</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c11">w</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:12_1_1_2_1"><span class="lab"><font color="Green" title="E10">A21</font></span></a>, <a class="ref" href="nbvectsp.html#D1" target="_self" onmouseover="rs('nbvectsp/D1')" onmouseout="rh()">Def1</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:12_1_1_2"/>
 <a href="vectsp_1.html#G1" title="VECTSP_1:aggr.1">ModuleStr</a>(# <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K1" title="NBVECTSP:func.1">XORB</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K2" title="NBVECTSP:func.2">ZeroB</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K4" title="NBVECTSP:func.4">MLTB</a> <font color="Maroon" title="c1">n</font></span>)</span> #) is  <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D3" onmouseover="rs('rlvect_1/D3')" onmouseout="rh()">RLVECT_1:def 3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E4:12_1_1"/><span class="lab"><font color="Green" title="E9">A22</font></span>: 
 <a href="vectsp_1.html#G1" title="VECTSP_1:aggr.1">ModuleStr</a>(# <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K1" title="NBVECTSP:func.1">XORB</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K2" title="NBVECTSP:func.2">ZeroB</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K4" title="NBVECTSP:func.4">MLTB</a> <font color="Maroon" title="c1">n</font></span>)</span> #) is  <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="12_1_1_3"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:12_1_1_3"/>
 for <font color="Olive" title="b1">v</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="vectsp_1.html#G1" title="VECTSP_1:aggr.1">ModuleStr</a>(# <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K1" title="NBVECTSP:func.1">XORB</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K2" title="NBVECTSP:func.2">ZeroB</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K4" title="NBVECTSP:func.4">MLTB</a> <font color="Maroon" title="c1">n</font></span>)</span> #) holds  <font color="Olive" title="b1">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="vectsp_1.html#G1" title="VECTSP_1:aggr.1">ModuleStr</a>(# <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K1" title="NBVECTSP:func.1">XORB</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K2" title="NBVECTSP:func.2">ZeroB</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K4" title="NBVECTSP:func.4">MLTB</a> <font color="Maroon" title="c1">n</font></span>)</span> #)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="12_1_1_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">v</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="vectsp_1.html#G1" title="VECTSP_1:aggr.1">ModuleStr</a>(# <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K1" title="NBVECTSP:func.1">XORB</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K2" title="NBVECTSP:func.2">ZeroB</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K4" title="NBVECTSP:func.4">MLTB</a> <font color="Maroon" title="c1">n</font></span>)</span> #); <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="vectsp_1.html#G1" title="VECTSP_1:aggr.1">ModuleStr</a>(# <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K1" title="NBVECTSP:func.1">XORB</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K2" title="NBVECTSP:func.2">ZeroB</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K4" title="NBVECTSP:func.4">MLTB</a> <font color="Maroon" title="c1">n</font></span>)</span> #)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">v</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c10">v1</font> = <font color="Maroon" title="c9">v</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a> ;<br/>
<span class="kw">thus </span><font color="Maroon" title="c9">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="vectsp_1.html#G1" title="VECTSP_1:aggr.1">ModuleStr</a>(# <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K1" title="NBVECTSP:func.1">XORB</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K2" title="NBVECTSP:func.2">ZeroB</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K4" title="NBVECTSP:func.4">MLTB</a> <font color="Maroon" title="c1">n</font></span>)</span> #)</span>)</span> = 
 <a href="descip_1.html#K11" title="DESCIP_1:func.11">Op-XOR</a> (<font color="Maroon" title="c10">v1</font>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K2" title="NBVECTSP:func.2">ZeroB</a> <font color="Maroon" title="c1">n</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="nbvectsp.html#D1" target="_self" onmouseover="rs('nbvectsp/D1')" onmouseout="rh()">Def1</a></span>
<br/>.= 
<font color="Maroon" title="c9">v</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nbvectsp.html#T2" target="_self" onmouseover="rs('nbvectsp/T2')" onmouseout="rh()">Th2</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:12_1_1_3"/>
 <a href="vectsp_1.html#G1" title="VECTSP_1:aggr.1">ModuleStr</a>(# <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K1" title="NBVECTSP:func.1">XORB</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K2" title="NBVECTSP:func.2">ZeroB</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K4" title="NBVECTSP:func.4">MLTB</a> <font color="Maroon" title="c1">n</font></span>)</span> #) is  <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D4" onmouseover="rs('rlvect_1/D4')" onmouseout="rh()">RLVECT_1:def 4</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E5:12_1_1"/><span class="lab"><font color="Green" title="E10">A23</font></span>: 
 <a href="vectsp_1.html#G1" title="VECTSP_1:aggr.1">ModuleStr</a>(# <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K1" title="NBVECTSP:func.1">XORB</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K2" title="NBVECTSP:func.2">ZeroB</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K4" title="NBVECTSP:func.4">MLTB</a> <font color="Maroon" title="c1">n</font></span>)</span> #) is  <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="12_1_1_4"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:12_1_1_4"/>
 for <font color="Olive" title="b1">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="vectsp_1.html#G1" title="VECTSP_1:aggr.1">ModuleStr</a>(# <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K1" title="NBVECTSP:func.1">XORB</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K2" title="NBVECTSP:func.2">ZeroB</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K4" title="NBVECTSP:func.4">MLTB</a> <font color="Maroon" title="c1">n</font></span>)</span> #) holds  <font color="Olive" title="b1">x</font> is  <a href="algstr_0.html#V10" title="ALGSTR_0:attr.10">right_complementable</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="12_1_1_4_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">v</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="vectsp_1.html#G1" title="VECTSP_1:aggr.1">ModuleStr</a>(# <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K1" title="NBVECTSP:func.1">XORB</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K2" title="NBVECTSP:func.2">ZeroB</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K4" title="NBVECTSP:func.4">MLTB</a> <font color="Maroon" title="c1">n</font></span>)</span> #); <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">v</font> is  <a href="algstr_0.html#V10" title="ALGSTR_0:attr.10">right_complementable</a> </span><br/>

<a NAME="E1:12_1_1_4_1"/>
 ex <font color="Olive" title="b1">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="vectsp_1.html#G1" title="VECTSP_1:aggr.1">ModuleStr</a>(# <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K1" title="NBVECTSP:func.1">XORB</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K2" title="NBVECTSP:func.2">ZeroB</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K4" title="NBVECTSP:func.4">MLTB</a> <font color="Maroon" title="c1">n</font></span>)</span> #) st <font color="Maroon" title="c9">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b1">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> <a href="vectsp_1.html#G1" title="VECTSP_1:aggr.1">ModuleStr</a>(# <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K1" title="NBVECTSP:func.1">XORB</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K2" title="NBVECTSP:func.2">ZeroB</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K4" title="NBVECTSP:func.4">MLTB</a> <font color="Maroon" title="c1">n</font></span>)</span> #)
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="12_1_1_4_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c10">v1</font> = <font color="Maroon" title="c9">v</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a> ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c9">v</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c9">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="vectsp_1.html#G1" title="VECTSP_1:aggr.1">ModuleStr</a>(# <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K1" title="NBVECTSP:func.1">XORB</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K2" title="NBVECTSP:func.2">ZeroB</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K4" title="NBVECTSP:func.4">MLTB</a> <font color="Maroon" title="c1">n</font></span>)</span> #)</span><br/>

<span class="kw">thus </span><font color="Maroon" title="c9">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c9">v</font> = 
 <a href="descip_1.html#K11" title="DESCIP_1:func.11">Op-XOR</a> (<font color="Maroon" title="c10">v1</font>,<font color="Maroon" title="c10">v1</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="nbvectsp.html#D1" target="_self" onmouseover="rs('nbvectsp/D1')" onmouseout="rh()">Def1</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="vectsp_1.html#G1" title="VECTSP_1:aggr.1">ModuleStr</a>(# <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K1" title="NBVECTSP:func.1">XORB</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K2" title="NBVECTSP:func.2">ZeroB</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K4" title="NBVECTSP:func.4">MLTB</a> <font color="Maroon" title="c1">n</font></span>)</span> #)
<span class="kw">by</span> <span class="lab"><a class="ref" href="nbvectsp.html#T3" target="_self" onmouseover="rs('nbvectsp/T3')" onmouseout="rh()">Th3</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:12_1_1_4_1"/>
<font color="Maroon" title="c9">v</font> is  <a href="algstr_0.html#V10" title="ALGSTR_0:attr.10">right_complementable</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="algstr_0.html#D11" onmouseover="rs('algstr_0/D11')" onmouseout="rh()">ALGSTR_0:def 11</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:12_1_1_4"/>
 <a href="vectsp_1.html#G1" title="VECTSP_1:aggr.1">ModuleStr</a>(# <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K1" title="NBVECTSP:func.1">XORB</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K2" title="NBVECTSP:func.2">ZeroB</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K4" title="NBVECTSP:func.4">MLTB</a> <font color="Maroon" title="c1">n</font></span>)</span> #) is  <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="algstr_0.html#D16" onmouseover="rs('algstr_0/D16')" onmouseout="rh()">ALGSTR_0:def 16</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E6:12_1_1"/>
 <a href="vectsp_1.html#G1" title="VECTSP_1:aggr.1">ModuleStr</a>(# <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K1" title="NBVECTSP:func.1">XORB</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K2" title="NBVECTSP:func.2">ZeroB</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K4" title="NBVECTSP:func.4">MLTB</a> <font color="Maroon" title="c1">n</font></span>)</span> #) is  <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="12_1_1_5"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:12_1_1_5"/>
 for <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="vectsp_1.html#G1" title="VECTSP_1:aggr.1">ModuleStr</a>(# <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K1" title="NBVECTSP:func.1">XORB</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K2" title="NBVECTSP:func.2">ZeroB</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K4" title="NBVECTSP:func.4">MLTB</a> <font color="Maroon" title="c1">n</font></span>)</span> #) holds  <font color="Olive" title="b1">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b2">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">w</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b1">v</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="12_1_1_5_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">u</font>, <font color="Maroon" title="c10">v</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="vectsp_1.html#G1" title="VECTSP_1:aggr.1">ModuleStr</a>(# <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K1" title="NBVECTSP:func.1">XORB</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K2" title="NBVECTSP:func.2">ZeroB</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K4" title="NBVECTSP:func.4">MLTB</a> <font color="Maroon" title="c1">n</font></span>)</span> #); <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">u</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c10">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c9">u</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c11">u1</font> = <font color="Maroon" title="c9">u</font>, <font color="Maroon" title="c12">v1</font> = <font color="Maroon" title="c10">v</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a> ;<br/>
<span class="kw">thus </span><font color="Maroon" title="c9">u</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c10">v</font> = 
 <a href="descip_1.html#K11" title="DESCIP_1:func.11">Op-XOR</a> (<font color="Maroon" title="c12">v1</font>,<font color="Maroon" title="c11">u1</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="nbvectsp.html#D1" target="_self" onmouseover="rs('nbvectsp/D1')" onmouseout="rh()">Def1</a></span>
<br/>.= 
<font color="Maroon" title="c10">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c9">u</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nbvectsp.html#D1" target="_self" onmouseover="rs('nbvectsp/D1')" onmouseout="rh()">Def1</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:12_1_1_5"/>
 <a href="vectsp_1.html#G1" title="VECTSP_1:aggr.1">ModuleStr</a>(# <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K1" title="NBVECTSP:func.1">XORB</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K2" title="NBVECTSP:func.2">ZeroB</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K4" title="NBVECTSP:func.4">MLTB</a> <font color="Maroon" title="c1">n</font></span>)</span> #) is  <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D2" onmouseover="rs('rlvect_1/D2')" onmouseout="rh()">RLVECT_1:def 2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E7:12_1_1"/>
 <a href="vectsp_1.html#G1" title="VECTSP_1:aggr.1">ModuleStr</a>(# <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K1" title="NBVECTSP:func.1">XORB</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K2" title="NBVECTSP:func.2">ZeroB</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="nbvectsp.html#K4" title="NBVECTSP:func.4">MLTB</a> <font color="Maroon" title="c1">n</font></span>)</span> #) is   <a href="vectsp_1.html#NM9" title="VECTSP_1:NM.9">VectSp</a> of  <a href="bspace.html#K2" title="BSPACE:func.2">Z_2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:12_1_1"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="txt" href="#E3:12_1_1"><span class="lab"><font color="Green" title="E8">A19</font></span></a>, <a class="txt" href="#E4:12_1_1"><span class="lab"><font color="Green" title="E9">A22</font></span></a>, <a class="txt" href="#E5:12_1_1"><span class="lab"><font color="Green" title="E10">A23</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
