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

<b>let </b><font color="Maroon" title="c1">UN</font> be   <a href="classes2.html#NM1" title="CLASSES2:NM.1">Universe</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">UN</font> &amp;  <a href="ringcat1.html#R2" title="RINGCAT1:pred.2">GO</a> <font color="Olive" title="b1">x</font>, <a href="mod_2.html#K18" title="MOD_2:func.18">Z3</a>  )</span><br/>

<b>set </b><font color="Maroon" title="c2">G</font> =  <a href="mod_2.html#K18" title="MOD_2:func.18">Z3</a> ;<br/>
<b>reconsider </b><font color="Maroon" title="c3">x1</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="mod_2.html#K18" title="MOD_2:func.18">Z3</a> , <font color="Maroon" title="c4">x2</font> = the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <a href="mod_2.html#K18" title="MOD_2:func.18">Z3</a> , <font color="Maroon" title="c5">x3</font> =  <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">comp</a> <a href="mod_2.html#K18" title="MOD_2:func.18">Z3</a> , <font color="Maroon" title="c6">x4</font> =  <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> , <font color="Maroon" title="c7">x5</font> = the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <a href="mod_2.html#K18" title="MOD_2:func.18">Z3</a> , <font color="Maroon" title="c8">x6</font> =  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <a href="mod_2.html#K18" title="MOD_2:func.18">Z3</a>  as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">UN</font> <b>by </b><i><a class="ref" href="mod_2.html#T40" title="MOD_2:th.40">MOD_2:40</a></i>;<br/>
<b>set </b><font color="Maroon" title="c9">x'</font> = <span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">x3</font>,<font color="Maroon" title="c6">x4</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span>;<br/>
<b>set </b><font color="Maroon" title="c10">x</font> = <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><span class="p2"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">x3</font>,<font color="Maroon" title="c6">x4</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span>,<font color="Maroon" title="c7">x5</font>,<font color="Maroon" title="c8">x6</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span>;<br/>
<a NAME="E2:55"/><i><font color="Green" title="E34">A1</font></i>: 
<span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">x3</font>,<font color="Maroon" title="c6">x4</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">UN</font>
 
<b>by </b><i><a class="ref" href="grcat_1.html#T3" title="GRCAT_1:th.3">GRCAT_1:3</a></i>;<br/>
<a NAME="E3:55"/><i><font color="Green" title="E35">A2</font></i>: 
 <a href="ringcat1.html#R2" title="RINGCAT1:pred.2">GO</a> <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><span class="p2"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">x3</font>,<font color="Maroon" title="c6">x4</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span>,<font color="Maroon" title="c7">x5</font>,<font color="Maroon" title="c8">x6</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span>, <a href="mod_2.html#K18" title="MOD_2:func.18">Z3</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon" title="c3">x1</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="ringcat1.html#D16" target="_self" title="RINGCAT1:def.16">RINGCAT1:def 16</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">x2</font>, <font color="Olive" title="b2">x3</font>, <font color="Olive" title="b3">x4</font>, <font color="Olive" title="b4">x5</font>, <font color="Olive" title="b5">x6</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><span class="p2"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">x3</font>,<font color="Maroon" title="c6">x4</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span>,<font color="Maroon" title="c7">x5</font>,<font color="Maroon" title="c8">x6</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><span class="p2"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Maroon" title="c3">x1</font>,<font color="Olive" title="b1">x2</font>,<font color="Olive" title="b2">x3</font>,<font color="Olive" title="b3">x4</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span>,<font color="Olive" title="b4">x5</font>,<font color="Olive" title="b5">x6</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span> &amp;  ex <font color="Olive" title="b6">G</font> being <a href="algstr_0.html#V36" title="ALGSTR_0:attr.36">strict</a> <a href="funcsdom.html#NM1" title="FUNCSDOM:NM.1">Ring</a> st <br/>(  <a href="mod_2.html#K18" title="MOD_2:func.18">Z3</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">G</font> &amp; <font color="Maroon" title="c3">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b6">G</font> &amp; <font color="Olive" title="b1">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Olive" title="b6">G</font> &amp; <font color="Olive" title="b2">x3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">comp</a> <font color="Olive" title="b6">G</font> &amp; <font color="Olive" title="b3">x4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Olive" title="b6">G</font> &amp; <font color="Olive" title="b4">x5</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Olive" title="b6">G</font> &amp; <font color="Olive" title="b5">x6</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Olive" title="b6">G</font> ) )</span><br/>

<b>take </b>
<font color="Maroon" title="c4">x2</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">x3</font>, <font color="Olive" title="b2">x4</font>, <font color="Olive" title="b3">x5</font>, <font color="Olive" title="b4">x6</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><span class="p2"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">x3</font>,<font color="Maroon" title="c6">x4</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span>,<font color="Maroon" title="c7">x5</font>,<font color="Maroon" title="c8">x6</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><span class="p2"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c4">x2</font>,<font color="Olive" title="b1">x3</font>,<font color="Olive" title="b2">x4</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span>,<font color="Olive" title="b3">x5</font>,<font color="Olive" title="b4">x6</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span> &amp;  ex <font color="Olive" title="b5">G</font> being <a href="algstr_0.html#V36" title="ALGSTR_0:attr.36">strict</a> <a href="funcsdom.html#NM1" title="FUNCSDOM:NM.1">Ring</a> st <br/>(  <a href="mod_2.html#K18" title="MOD_2:func.18">Z3</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">G</font> &amp; <font color="Maroon" title="c3">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b5">G</font> &amp; <font color="Maroon" title="c4">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Olive" title="b5">G</font> &amp; <font color="Olive" title="b1">x3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">comp</a> <font color="Olive" title="b5">G</font> &amp; <font color="Olive" title="b2">x4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Olive" title="b5">G</font> &amp; <font color="Olive" title="b3">x5</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Olive" title="b5">G</font> &amp; <font color="Olive" title="b4">x6</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Olive" title="b5">G</font> ) )</span><br/>

<b>take </b>
<font color="Maroon" title="c5">x3</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">x4</font>, <font color="Olive" title="b2">x5</font>, <font color="Olive" title="b3">x6</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><span class="p2"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">x3</font>,<font color="Maroon" title="c6">x4</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span>,<font color="Maroon" title="c7">x5</font>,<font color="Maroon" title="c8">x6</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><span class="p2"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">x3</font>,<font color="Olive" title="b1">x4</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span>,<font color="Olive" title="b2">x5</font>,<font color="Olive" title="b3">x6</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span> &amp;  ex <font color="Olive" title="b4">G</font> being <a href="algstr_0.html#V36" title="ALGSTR_0:attr.36">strict</a> <a href="funcsdom.html#NM1" title="FUNCSDOM:NM.1">Ring</a> st <br/>(  <a href="mod_2.html#K18" title="MOD_2:func.18">Z3</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">G</font> &amp; <font color="Maroon" title="c3">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b4">G</font> &amp; <font color="Maroon" title="c4">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Olive" title="b4">G</font> &amp; <font color="Maroon" title="c5">x3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">comp</a> <font color="Olive" title="b4">G</font> &amp; <font color="Olive" title="b1">x4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Olive" title="b4">G</font> &amp; <font color="Olive" title="b2">x5</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Olive" title="b4">G</font> &amp; <font color="Olive" title="b3">x6</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Olive" title="b4">G</font> ) )</span><br/>

<b>take </b>
<font color="Maroon" title="c6">x4</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">x5</font>, <font color="Olive" title="b2">x6</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><span class="p2"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">x3</font>,<font color="Maroon" title="c6">x4</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span>,<font color="Maroon" title="c7">x5</font>,<font color="Maroon" title="c8">x6</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><span class="p2"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">x3</font>,<font color="Maroon" title="c6">x4</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span>,<font color="Olive" title="b1">x5</font>,<font color="Olive" title="b2">x6</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span> &amp;  ex <font color="Olive" title="b3">G</font> being <a href="algstr_0.html#V36" title="ALGSTR_0:attr.36">strict</a> <a href="funcsdom.html#NM1" title="FUNCSDOM:NM.1">Ring</a> st <br/>(  <a href="mod_2.html#K18" title="MOD_2:func.18">Z3</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">G</font> &amp; <font color="Maroon" title="c3">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b3">G</font> &amp; <font color="Maroon" title="c4">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Olive" title="b3">G</font> &amp; <font color="Maroon" title="c5">x3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">comp</a> <font color="Olive" title="b3">G</font> &amp; <font color="Maroon" title="c6">x4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Olive" title="b3">G</font> &amp; <font color="Olive" title="b1">x5</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Olive" title="b3">G</font> &amp; <font color="Olive" title="b2">x6</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Olive" title="b3">G</font> ) )</span><br/>

<b>take </b>
<font color="Maroon" title="c7">x5</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">x6</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><span class="p2"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">x3</font>,<font color="Maroon" title="c6">x4</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span>,<font color="Maroon" title="c7">x5</font>,<font color="Maroon" title="c8">x6</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><span class="p2"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">x3</font>,<font color="Maroon" title="c6">x4</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span>,<font color="Maroon" title="c7">x5</font>,<font color="Olive" title="b1">x6</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span> &amp;  ex <font color="Olive" title="b2">G</font> being <a href="algstr_0.html#V36" title="ALGSTR_0:attr.36">strict</a> <a href="funcsdom.html#NM1" title="FUNCSDOM:NM.1">Ring</a> st <br/>(  <a href="mod_2.html#K18" title="MOD_2:func.18">Z3</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">G</font> &amp; <font color="Maroon" title="c3">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b2">G</font> &amp; <font color="Maroon" title="c4">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Olive" title="b2">G</font> &amp; <font color="Maroon" title="c5">x3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">comp</a> <font color="Olive" title="b2">G</font> &amp; <font color="Maroon" title="c6">x4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Olive" title="b2">G</font> &amp; <font color="Maroon" title="c7">x5</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Olive" title="b2">G</font> &amp; <font color="Olive" title="b1">x6</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Olive" title="b2">G</font> ) )</span><br/>

<b>take </b>
<font color="Maroon" title="c8">x6</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><span class="p2"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">x3</font>,<font color="Maroon" title="c6">x4</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span>,<font color="Maroon" title="c7">x5</font>,<font color="Maroon" title="c8">x6</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><span class="p2"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">x3</font>,<font color="Maroon" title="c6">x4</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span>,<font color="Maroon" title="c7">x5</font>,<font color="Maroon" title="c8">x6</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span> &amp;  ex <font color="Olive" title="b1">G</font> being <a href="algstr_0.html#V36" title="ALGSTR_0:attr.36">strict</a> <a href="funcsdom.html#NM1" title="FUNCSDOM:NM.1">Ring</a> st <br/>(  <a href="mod_2.html#K18" title="MOD_2:func.18">Z3</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">G</font> &amp; <font color="Maroon" title="c3">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">G</font> &amp; <font color="Maroon" title="c4">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Olive" title="b1">G</font> &amp; <font color="Maroon" title="c5">x3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">comp</a> <font color="Olive" title="b1">G</font> &amp; <font color="Maroon" title="c6">x4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Olive" title="b1">G</font> &amp; <font color="Maroon" title="c7">x5</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Olive" title="b1">G</font> &amp; <font color="Maroon" title="c8">x6</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Olive" title="b1">G</font> ) )</span><br/>

<b>thus </b><a NAME="E1:55_1"/>
( <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><span class="p2"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">x3</font>,<font color="Maroon" title="c6">x4</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span>,<font color="Maroon" title="c7">x5</font>,<font color="Maroon" title="c8">x6</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><span class="p2"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">x3</font>,<font color="Maroon" title="c6">x4</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span>,<font color="Maroon" title="c7">x5</font>,<font color="Maroon" title="c8">x6</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span> &amp;  ex <font color="Olive" title="b1">G</font> being <a href="algstr_0.html#V36" title="ALGSTR_0:attr.36">strict</a> <a href="funcsdom.html#NM1" title="FUNCSDOM:NM.1">Ring</a> st <br/>(  <a href="mod_2.html#K18" title="MOD_2:func.18">Z3</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">G</font> &amp; <font color="Maroon" title="c3">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">G</font> &amp; <font color="Maroon" title="c4">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Olive" title="b1">G</font> &amp; <font color="Maroon" title="c5">x3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">comp</a> <font color="Olive" title="b1">G</font> &amp; <font color="Maroon" title="c6">x4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Olive" title="b1">G</font> &amp; <font color="Maroon" title="c7">x5</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Olive" title="b1">G</font> &amp; <font color="Maroon" title="c8">x6</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Olive" title="b1">G</font> ) )
 ; <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>take </b>
<span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><span class="p2"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">x3</font>,<font color="Maroon" title="c6">x4</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span>,<font color="Maroon" title="c7">x5</font>,<font color="Maroon" title="c8">x6</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><span class="p2"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">x3</font>,<font color="Maroon" title="c6">x4</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span>,<font color="Maroon" title="c7">x5</font>,<font color="Maroon" title="c8">x6</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">UN</font> &amp;  <a href="ringcat1.html#R2" title="RINGCAT1:pred.2">GO</a> <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><span class="p2"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">x3</font>,<font color="Maroon" title="c6">x4</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span>,<font color="Maroon" title="c7">x5</font>,<font color="Maroon" title="c8">x6</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span>, <a href="mod_2.html#K18" title="MOD_2:func.18">Z3</a>  )</span><br/>

<b>thus </b><a NAME="E4:55"/>
( <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><span class="p2"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">x3</font>,<font color="Maroon" title="c6">x4</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span>,<font color="Maroon" title="c7">x5</font>,<font color="Maroon" title="c8">x6</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">UN</font> &amp;  <a href="ringcat1.html#R2" title="RINGCAT1:pred.2">GO</a> <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><span class="p2"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">x3</font>,<font color="Maroon" title="c6">x4</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span>,<font color="Maroon" title="c7">x5</font>,<font color="Maroon" title="c8">x6</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span>, <a href="mod_2.html#K18" title="MOD_2:func.18">Z3</a>  )
 <b>by </b><i><a class="txt" href="ringcat1.html#E2:55"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="ringcat1.html#E3:55"><i><font color="Green" title="E35">A2</font></i></a>, <a class="ref" href="grcat_1.html#T3" title="GRCAT_1:th.3">GRCAT_1:3</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>
