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

<b>thus </b><a NAME="E1:51_1_1"/>
 <a href="mod_2.html#K18" title="MOD_2:func.18">Z3</a>  is <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="mod_2.html#K18" title="MOD_2:func.18">Z3</a>  is <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a> &amp;  <a href="mod_2.html#K18" title="MOD_2:func.18">Z3</a>  is <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c2">y</font>, <font color="Maroon" title="c3">z</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="mod_2.html#K18" title="MOD_2:func.18">Z3</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c3">z</font></span>)</span></span><br/>

<b>reconsider </b><font color="Maroon" title="c4">X</font> = <font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c5">Y</font> = <font color="Maroon" title="c2">y</font>, <font color="Maroon" title="c6">Z</font> = <font color="Maroon" title="c3">z</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,2</span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> ;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c3">z</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c5">Y</font></span>)</span> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c6">Z</font>
<b>by </b><i><a class="txt" href="mod_2.html#E61"><i><font color="Green" title="E36">Lm6</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c4">X</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Y</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c6">Z</font></span>)</span>
<b>by </b><i><a class="txt" href="mod_2.html#E62"><i><font color="Green" title="E37">Lm7</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c1">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c3">z</font></span>)</span>
<b>by </b><i><a class="txt" href="mod_2.html#E61"><i><font color="Green" title="E36">Lm6</font></i></a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:51_1_1"/>
 <a href="mod_2.html#K18" title="MOD_2:func.18">Z3</a>  is <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="mod_2.html#K18" title="MOD_2:func.18">Z3</a>  is <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c1">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="mod_2.html#K18" title="MOD_2:func.18">Z3</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D7" title="RLVECT_1:def.7">RLVECT_1:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">x</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="mod_2.html#K18" title="MOD_2:func.18">Z3</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c2">X</font> = <font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c3">a</font> =  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,2</span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> <b>by </b><i><a class="ref" href="mod_2.html#D23" target="_self" title="MOD_2:def.23">Def23</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c1">x</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="mod_2.html#K18" title="MOD_2:func.18">Z3</a> </span>)</span> = 
<font color="Maroon" title="c2">X</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c3">a</font>
<b>by </b><i><a class="ref" href="mod_2.html#D18" target="_self" title="MOD_2:def.18">Def18</a></i>
<br/>.= 
<font color="Maroon" title="c1">x</font>
<b>by </b><i><a class="txt" href="mod_2.html#E62"><i><font color="Green" title="E37">Lm7</font></i></a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon" title="c1">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="mod_2.html#K18" title="MOD_2:func.18">Z3</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="algstr_0.html#D16" title="ALGSTR_0:def.16">ALGSTR_0:def 16</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">x</font> is <a href="algstr_0.html#V10" title="ALGSTR_0:attr.10">right_complementable</a></span><br/>

<b>reconsider </b><font color="Maroon" title="c2">x'</font> = <font color="Maroon" title="c1">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,2</span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> ;<br/>
<b>reconsider </b><font color="Maroon" title="c3">y</font> = <a href="mod_2.html#K15" title="MOD_2:func.15">compl3</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">x'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="mod_2.html#K18" title="MOD_2:func.18">Z3</a>  ;<br/>
<b>reconsider </b><font color="Maroon" title="c4">y'</font> = <font color="Maroon" title="c3">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,2</span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> ;<br/>
<b>take </b>
<font color="Maroon" title="c3">y</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="algstr_0.html#D11" title="ALGSTR_0:def.11">ALGSTR_0:def 11</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c3">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="mod_2.html#K18" title="MOD_2:func.18">Z3</a> </span><br/>

<a NAME="E6:51_1_1"/><i><font color="Green" title="E38">A1</font></i>: 
<font color="Maroon" title="c4">y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mod_2.html#K10" title="MOD_2:func.10">-</a> <font color="Maroon" title="c2">x'</font>
 
<b>by </b><i><a class="ref" href="mod_2.html#D20" target="_self" title="MOD_2:def.20">Def20</a></i>;<br/>
<b>thus </b><a NAME="E7:51_1_1"/>
<font color="Maroon" title="c1">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c3">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="mod_2.html#K18" title="MOD_2:func.18">Z3</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_1_3"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:51_1_1_3_1"/>
( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 </span><b>by </b><i><a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_3_1_1"><b>suppose </b></a><a NAME="E1:51_1_1_3_1_1"/><i><font color="Green" title="E39">A2</font></i>: 
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c3">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="mod_2.html#K18" title="MOD_2:func.18">Z3</a> </span><br/><div class="add"><a NAME="E2:51_1_1_3_1_1"/><b>then </b><i><font color="Green" title="E40">A3</font></i>: 
<font color="Maroon" title="c4">y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="mod_2.html#E6:51_1_1"><i><font color="Green" title="E38">A1</font></i></a>, <a class="ref" href="mod_2.html#D15" target="_self" title="MOD_2:def.15">Def15</a></i>;<br/><b>thus </b><font color="Maroon" title="c1">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c3">y</font> = 
<font color="Maroon" title="c2">x'</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c4">y'</font>
<b>by </b><i><a class="ref" href="mod_2.html#D18" target="_self" title="MOD_2:def.18">Def18</a></i>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="mod_2.html#K18" title="MOD_2:func.18">Z3</a> 
<b>by </b><i><a class="txt" href="mod_2.html#E1:51_1_1_3_1_1"><i><font color="Green" title="E39">A2</font></i></a>, <a class="txt" href="mod_2.html#E2:51_1_1_3_1_1"><i><font color="Green" title="E40">A3</font></i></a>, <a class="ref" href="mod_2.html#D16" target="_self" title="MOD_2:def.16">Def16</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_3_1_2"><b>suppose </b></a><a NAME="E1:51_1_1_3_1_2"/><i><font color="Green" title="E39">A4</font></i>: 
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c3">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="mod_2.html#K18" title="MOD_2:func.18">Z3</a> </span><br/><div class="add"><a NAME="E2:51_1_1_3_1_2"/><b>then </b><i><font color="Green" title="E40">A5</font></i>: 
<font color="Maroon" title="c4">y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 <b>by </b><i><a class="txt" href="mod_2.html#E6:51_1_1"><i><font color="Green" title="E38">A1</font></i></a>, <a class="ref" href="mod_2.html#D15" target="_self" title="MOD_2:def.15">Def15</a></i>;<br/><b>thus </b><font color="Maroon" title="c1">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c3">y</font> = 
<font color="Maroon" title="c2">x'</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c4">y'</font>
<b>by </b><i><a class="ref" href="mod_2.html#D18" target="_self" title="MOD_2:def.18">Def18</a></i>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="mod_2.html#K18" title="MOD_2:func.18">Z3</a> 
<b>by </b><i><a class="txt" href="mod_2.html#E1:51_1_1_3_1_2"><i><font color="Green" title="E39">A4</font></i></a>, <a class="txt" href="mod_2.html#E2:51_1_1_3_1_2"><i><font color="Green" title="E40">A5</font></i></a>, <a class="ref" href="mod_2.html#D16" target="_self" title="MOD_2:def.16">Def16</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_3_1_3"><b>suppose </b></a><a NAME="E1:51_1_1_3_1_3"/><i><font color="Green" title="E39">A6</font></i>: 
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c3">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="mod_2.html#K18" title="MOD_2:func.18">Z3</a> </span><br/><div class="add"><a NAME="E2:51_1_1_3_1_3"/><b>then </b><i><font color="Green" title="E40">A7</font></i>: 
<font color="Maroon" title="c4">y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="mod_2.html#E6:51_1_1"><i><font color="Green" title="E38">A1</font></i></a>, <a class="ref" href="mod_2.html#D15" target="_self" title="MOD_2:def.15">Def15</a></i>;<br/><b>thus </b><font color="Maroon" title="c1">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c3">y</font> = 
<font color="Maroon" title="c2">x'</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c4">y'</font>
<b>by </b><i><a class="ref" href="mod_2.html#D18" target="_self" title="MOD_2:def.18">Def18</a></i>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="mod_2.html#K18" title="MOD_2:func.18">Z3</a> 
<b>by </b><i><a class="txt" href="mod_2.html#E1:51_1_1_3_1_3"><i><font color="Green" title="E39">A6</font></i></a>, <a class="txt" href="mod_2.html#E2:51_1_1_3_1_3"><i><font color="Green" title="E40">A7</font></i></a>, <a class="ref" href="mod_2.html#D16" target="_self" title="MOD_2:def.16">Def16</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>


</div>
