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<div class="add">

<span class="kw">set </span><font color="Maroon" title="c2">IT</font> = <font color="Maroon" title="c1">n</font> <a href="nbvectsp.html#K3" title="NBVECTSP:func.3">-BinaryGroup</a> ;<br/>
<div><span class="kw">hereby </span> <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="rlvect_1.html#D3" onmouseover="rs('rlvect_1/D3')" onmouseout="rh()">RLVECT_1:def 3</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"> ( <font color="Maroon" title="c1">n</font> <a href="nbvectsp.html#K3" title="NBVECTSP:func.3">-BinaryGroup</a>  is  <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>  &amp; <font color="Maroon" title="c1">n</font> <a href="nbvectsp.html#K3" title="NBVECTSP:func.3">-BinaryGroup</a>  is  <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>  &amp; <font color="Maroon" title="c1">n</font> <a href="nbvectsp.html#K3" title="NBVECTSP:func.3">-BinaryGroup</a>  is  <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a>  &amp;  not <font color="Maroon" title="c1">n</font> <a href="nbvectsp.html#K3" title="NBVECTSP:func.3">-BinaryGroup</a>  is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  )</span>
<div class="add"><span class="kw">let </span><font color="Maroon" title="c3">u</font>, <font color="Maroon" title="c4">v</font>, <font color="Maroon" title="c5">w</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nbvectsp.html#K3" title="NBVECTSP:func.3">-BinaryGroup</a></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="c3">u</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c4">v</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">u</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">w</font></span>)</span></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c6">u1</font> = <font color="Maroon" title="c3">u</font>, <font color="Maroon" title="c7">v1</font> = <font color="Maroon" title="c4">v</font>, <font color="Maroon" title="c8">w1</font> = <font color="Maroon" title="c5">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="E2:8_1_1_1"/><span class="lab"><font color="Green" title="E6">A1</font></span>: 
<font color="Maroon" title="c3">u</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c4">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="c6">u1</font>,<font color="Maroon" title="c7">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="E3:8_1_1_1"/><span class="lab"><font color="Green" title="E7">A2</font></span>: 
<font color="Maroon" title="c4">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">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="c7">v1</font>,<font color="Maroon" title="c8">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="c3">u</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c4">v</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">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="c6">u1</font>,<font color="Maroon" title="c7">v1</font>)</span>)</span>,<font color="Maroon" title="c8">w1</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:8_1_1_1"><span class="lab"><font color="Green" title="E6">A1</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="c6">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="c7">v1</font>,<font color="Maroon" title="c8">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="c3">u</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">w</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:8_1_1_1"><span class="lab"><font color="Green" title="E7">A2</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>

<div><span class="kw">hereby </span> <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="rlvect_1.html#D4" onmouseover="rs('rlvect_1/D4')" onmouseout="rh()">RLVECT_1:def 4</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"> ( <font color="Maroon" title="c1">n</font> <a href="nbvectsp.html#K3" title="NBVECTSP:func.3">-BinaryGroup</a>  is  <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>  &amp; <font color="Maroon" title="c1">n</font> <a href="nbvectsp.html#K3" title="NBVECTSP:func.3">-BinaryGroup</a>  is  <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a>  &amp;  not <font color="Maroon" title="c1">n</font> <a href="nbvectsp.html#K3" title="NBVECTSP:func.3">-BinaryGroup</a>  is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  )</span>
<div class="add"><span class="kw">let </span><font color="Maroon" title="c3">v</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nbvectsp.html#K3" title="NBVECTSP:func.3">-BinaryGroup</a></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="c3">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> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nbvectsp.html#K3" title="NBVECTSP:func.3">-BinaryGroup</a></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v</font></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c4">v1</font> = <font color="Maroon" title="c3">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="c3">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> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nbvectsp.html#K3" title="NBVECTSP:func.3">-BinaryGroup</a></span>)</span></span>)</span> = 
 <a href="descip_1.html#K11" title="DESCIP_1:func.11">Op-XOR</a> (<font color="Maroon" title="c4">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="c3">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>

<div><span class="kw">hereby </span> <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="algstr_0.html#D16" onmouseover="rs('algstr_0/D16')" onmouseout="rh()">ALGSTR_0: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"> ( <font color="Maroon" title="c1">n</font> <a href="nbvectsp.html#K3" title="NBVECTSP:func.3">-BinaryGroup</a>  is  <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a>  &amp;  not <font color="Maroon" title="c1">n</font> <a href="nbvectsp.html#K3" title="NBVECTSP:func.3">-BinaryGroup</a>  is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  )</span>
<div class="add"><span class="kw">let </span><font color="Maroon" title="c3">v</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nbvectsp.html#K3" title="NBVECTSP:func.3">-BinaryGroup</a></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="c3">v</font> is  <a href="algstr_0.html#V10" title="ALGSTR_0:attr.10">right_complementable</a> </span><br/><span class="kw">thus </span><a NAME="E1:8_1_1_3"/>
<font color="Maroon" title="c3">v</font> is  <a href="algstr_0.html#V10" title="ALGSTR_0:attr.10">right_complementable</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="8_1_1_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c4">v1</font> = <font color="Maroon" title="c3">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="c3">v</font>
; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="algstr_0.html#D11" onmouseover="rs('algstr_0/D11')" onmouseout="rh()">ALGSTR_0:def 11</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"> <font color="Maroon" title="c3">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c3">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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nbvectsp.html#K3" title="NBVECTSP:func.3">-BinaryGroup</a></span>)</span></span><br/>

<span class="kw">thus </span><font color="Maroon" title="c3">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c3">v</font> = 
 <a href="descip_1.html#K11" title="DESCIP_1:func.11">Op-XOR</a> (<font color="Maroon" title="c4">v1</font>,<font color="Maroon" title="c4">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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nbvectsp.html#K3" title="NBVECTSP:func.3">-BinaryGroup</a></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></div>
<span class="kw">end;</span></div>

<div><span class="kw">hereby </span> <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="rlvect_1.html#D2" onmouseover="rs('rlvect_1/D2')" onmouseout="rh()">RLVECT_1:def 2</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">  not <font color="Maroon" title="c1">n</font> <a href="nbvectsp.html#K3" title="NBVECTSP:func.3">-BinaryGroup</a>  is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> </span>
<div class="add"><span class="kw">let </span><font color="Maroon" title="c3">u</font>, <font color="Maroon" title="c4">v</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nbvectsp.html#K3" title="NBVECTSP:func.3">-BinaryGroup</a></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="c3">u</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c4">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c3">u</font></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c5">u1</font> = <font color="Maroon" title="c3">u</font>, <font color="Maroon" title="c6">v1</font> = <font color="Maroon" title="c4">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="c3">u</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c4">v</font> = 
 <a href="descip_1.html#K11" title="DESCIP_1:func.11">Op-XOR</a> (<font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c5">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="c4">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c3">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">thus </span><a NAME="E5:8_1_1"/>
 not <font color="Maroon" title="c1">n</font> <a href="nbvectsp.html#K3" title="NBVECTSP:func.3">-BinaryGroup</a>  is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</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><br/>


</div>
