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

<b>set </b><font color="Maroon" title="c3">M</font> =  <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">LoopStr</a>(# <font color="Maroon" title="c2">I</font>,<span class="p1">(<span class="default"><a href="ideal_1.html#K1" title="IDEAL_1:func.1">add|</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c1">R</font></span>)</span>,<span class="p1">(<span class="default"><a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font></span>)</span>,<font color="Maroon" title="c2">I</font></span>)</span> #);<br/>
<b>reconsider </b><font color="Maroon" title="c4">M</font> =  <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">LoopStr</a>(# <font color="Maroon" title="c2">I</font>,<span class="p1">(<span class="default"><a href="ideal_1.html#K1" title="IDEAL_1:func.1">add|</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c1">R</font></span>)</span>,<span class="p1">(<span class="default"><a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font></span>)</span>,<font color="Maroon" title="c2">I</font></span>)</span> #) as  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="rlvect_1.html#L1" title="RLVECT_1:struct.1">LoopStr</a>  ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:37_1_1_1"/><i><font color="Green" title="E17">A1</font></i>: 
<font color="Maroon" title="c5">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c2">I</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>
 ;<br/><a NAME="E2:37_1_1_1"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c1">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><b>hence </b><a NAME="E3:37_1_1_1"/>
<font color="Maroon" title="c5">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="txt" href="ideal_1.html#E1:37_1_1_1"><i><font color="Green" title="E17">A1</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E3:37_1_1"/><b>then </b>
<span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c2">I</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c1">R</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E4:37_1_1"/><b>then </b><i><font color="Green" title="E17">A2</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default">the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c1">R</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c2">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c2">I</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></i>;<br/>
<a NAME="E5:37_1_1"/><i><font color="Green" title="E18">A3</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">M</font><br/> for <font color="Olive" title="b3">a'</font>, <font color="Olive" title="b4">b'</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">I</font>  st <font color="Olive" title="b3">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp; <font color="Olive" title="b4">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> holds <br/><font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">a'</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Olive" title="b4">b'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">M</font>;<br/><b>let </b><font color="Maroon" title="c7">a'</font>, <font color="Maroon" title="c8">b'</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">I</font>;<br/>



<a NAME="E1:37_1_1_2"/><i><font color="Green" title="E19">A4</font></i>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">a'</font>,<font color="Maroon" title="c8">b'</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default">the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c1">R</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c2">I</font></span>)</span>
 
<b>by </b><i><a class="txt" href="ideal_1.html#E4:37_1_1"><i><font color="Green" title="E17">A2</font></i></a></i>;<br/>
<b>assume </b><a NAME="E2:37_1_1_2"/>
( <font color="Maroon" title="c7">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> &amp; <font color="Maroon" title="c8">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> )
 ;<br/>

<b>hence </b><a NAME="E3:37_1_1_2"/>
<font color="Maroon" title="c5">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a'</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c8">b'</font>
 <b>by </b><i><a class="txt" href="ideal_1.html#E1:37_1_1_2"><i><font color="Green" title="E19">A4</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">b</font>, <font color="Maroon" title="c7">c</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">M</font>;<br/><b>reconsider </b><font color="Maroon" title="c8">a'</font> = <font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c9">b'</font> = <font color="Maroon" title="c6">b</font>, <font color="Maroon" title="c10">c'</font> = <font color="Maroon" title="c7">c</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">I</font> ;<br/><a NAME="E2:37_1_1_3"/>
<font color="Maroon" title="c8">a'</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c9">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font>
 <b>by </b><i><a class="ref" href="ideal_1.html#D1" target="_self" title="IDEAL_1:def.1">Def1</a></i>;<br/><a NAME="E3:37_1_1_3"/><b>then </b><i><font color="Green" title="E19">A5</font></i>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">a'</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c9">b'</font></span>)</span>,<font color="Maroon" title="c10">c'</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default">the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c1">R</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c2">I</font></span>)</span>
 <b>by </b><i><a class="txt" href="ideal_1.html#E4:37_1_1"><i><font color="Green" title="E17">A2</font></i></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/><a NAME="E4:37_1_1_3"/>
<font color="Maroon" title="c9">b'</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c10">c'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font>
 <b>by </b><i><a class="ref" href="ideal_1.html#D1" target="_self" title="IDEAL_1:def.1">Def1</a></i>;<br/><a NAME="E5:37_1_1_3"/><b>then </b><i><font color="Green" title="E20">A6</font></i>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c8">a'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">b'</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c10">c'</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default">the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c1">R</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c2">I</font></span>)</span>
 <b>by </b><i><a class="txt" href="ideal_1.html#E4:37_1_1"><i><font color="Green" title="E17">A2</font></i></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c7">c</font> = 
<span class="p1">(<span class="default">the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c1">R</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c2">I</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">a'</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c9">b'</font></span>)</span>,<font color="Maroon" title="c10">c'</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<b>by </b><i><a class="txt" href="ideal_1.html#E5:37_1_1"><i><font color="Green" title="E18">A3</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">a'</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c9">b'</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c10">c'</font>
<b>by </b><i><a class="txt" href="ideal_1.html#E3:37_1_1_3"><i><font color="Green" title="E19">A5</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>
<br/>.= 
<font color="Maroon" title="c8">a'</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">b'</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c10">c'</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="ideal_1.html#K1" title="IDEAL_1:func.1">add|</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c1">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c8">a'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">b'</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c10">c'</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<b>by </b><i><a class="txt" href="ideal_1.html#E5:37_1_1_3"><i><font color="Green" title="E20">A6</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>
<br/>.= 
<font color="Maroon" title="c5">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c7">c</font></span>)</span>
<b>by </b><i><a class="txt" href="ideal_1.html#E5:37_1_1"><i><font color="Green" title="E18">A3</font></i></a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:37_1_1"/>
 <a href="ideal_1.html#K3" title="IDEAL_1:func.3">Gr</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c1">R</font> is <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">add-associative</a>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></i>;<br/>


</div>
