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

<b>let </b><font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c2">y1</font>, <font color="Maroon" title="c3">y2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:54"/><i><font color="Green" title="E33">A1</font></i>: 
 <a href="ringcat1.html#R2" title="RINGCAT1:pred.2">GO</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y1</font>
 <b>and </b><br/><a NAME="E2:54"/><i><font color="Green" title="E34">A2</font></i>: 
 <a href="ringcat1.html#R2" title="RINGCAT1:pred.2">GO</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c3">y2</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c4">a1</font>, <font color="Maroon" title="c5">a2</font>, <font color="Maroon" title="c6">a3</font>, <font color="Maroon" title="c7">a4</font>, <font color="Maroon" title="c8">a5</font>, <font color="Maroon" title="c9">a6</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:54"/><i><font color="Green" title="E35">A3</font></i>: 
<font color="Maroon" title="c1">x</font> <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="c4">a1</font>,<font color="Maroon" title="c5">a2</font>,<font color="Maroon" title="c6">a3</font>,<font color="Maroon" title="c7">a4</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span>,<font color="Maroon" title="c8">a5</font>,<font color="Maroon" title="c9">a6</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span>
 <b>and </b><br/><a NAME="E5:54"/><i><font color="Green" title="E36">A4</font></i>: 
 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/>( <font color="Maroon" title="c2">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">G</font> &amp; <font color="Maroon" title="c4">a1</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="c5">a2</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="c6">a3</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="c7">a4</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="c8">a5</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="c9">a6</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> )
 <b>by </b><i><a class="txt" href="ringcat1.html#E1:54"><i><font color="Green" title="E33">A1</font></i></a>, <a class="ref" href="ringcat1.html#D16" target="_self" title="RINGCAT1:def.16">Def16</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">b1</font>, <font color="Maroon" title="c11">b2</font>, <font color="Maroon" title="c12">b3</font>, <font color="Maroon" title="c13">b4</font>, <font color="Maroon" title="c14">b5</font>, <font color="Maroon" title="c15">b6</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:54"/><i><font color="Green" title="E37">A5</font></i>: 
<font color="Maroon" title="c1">x</font> <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="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>,<font color="Maroon" title="c13">b4</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span>,<font color="Maroon" title="c14">b5</font>,<font color="Maroon" title="c15">b6</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span>
 <b>and </b><br/><a NAME="E8:54"/><i><font color="Green" title="E38">A6</font></i>: 
 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/>( <font color="Maroon" title="c3">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">G</font> &amp; <font color="Maroon" title="c10">b1</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="c11">b2</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="c12">b3</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="c13">b4</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="c14">b5</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="c15">b6</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> )
 <b>by </b><i><a class="txt" href="ringcat1.html#E2:54"><i><font color="Green" title="E34">A2</font></i></a>, <a class="ref" href="ringcat1.html#D16" target="_self" title="RINGCAT1:def.16">Def16</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c16">G1</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><b> such that </b><br/><a NAME="E10:54"/><i><font color="Green" title="E39">A7</font></i>: 
( <font color="Maroon" title="c2">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">G1</font> &amp; <font color="Maroon" title="c4">a1</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="Maroon" title="c16">G1</font> &amp; <font color="Maroon" title="c5">a2</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="Maroon" title="c16">G1</font> &amp; <font color="Maroon" title="c6">a3</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="Maroon" title="c16">G1</font> &amp; <font color="Maroon" title="c7">a4</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="Maroon" title="c16">G1</font> &amp; <font color="Maroon" title="c8">a5</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="Maroon" title="c16">G1</font> &amp; <font color="Maroon" title="c9">a6</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="Maroon" title="c16">G1</font> )
 <b>by </b><i><a class="txt" href="ringcat1.html#E5:54"><i><font color="Green" title="E36">A4</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c17">G2</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><b> such that </b><br/><a NAME="E12:54"/><i><font color="Green" title="E40">A8</font></i>: 
( <font color="Maroon" title="c3">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">G2</font> &amp; <font color="Maroon" title="c10">b1</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="Maroon" title="c17">G2</font> &amp; <font color="Maroon" title="c11">b2</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="Maroon" title="c17">G2</font> &amp; <font color="Maroon" title="c12">b3</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="Maroon" title="c17">G2</font> &amp; <font color="Maroon" title="c13">b4</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="Maroon" title="c17">G2</font> &amp; <font color="Maroon" title="c14">b5</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="Maroon" title="c17">G2</font> &amp; <font color="Maroon" title="c15">b6</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="Maroon" title="c17">G2</font> )
 <b>by </b><i><a class="txt" href="ringcat1.html#E8:54"><i><font color="Green" title="E38">A6</font></i></a></i>;<br/>
<a NAME="E13:54"/>
( <span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Maroon" title="c4">a1</font>,<font color="Maroon" title="c5">a2</font>,<font color="Maroon" title="c6">a3</font>,<font color="Maroon" title="c7">a4</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>,<font color="Maroon" title="c13">b4</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> &amp; <font color="Maroon" title="c8">a5</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">b5</font> &amp; <font color="Maroon" title="c9">a6</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">b6</font> )
 
<b>by </b><i><a class="txt" href="ringcat1.html#E4:54"><i><font color="Green" title="E35">A3</font></i></a>, <a class="txt" href="ringcat1.html#E7:54"><i><font color="Green" title="E37">A5</font></i></a>, <a class="ref" href="mcart_1.html#T28" title="MCART_1:th.28">MCART_1:28</a></i>;<br/>
<a NAME="E14:54"/><b>then </b>
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c16">G1</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="Maroon" title="c17">G2</font> &amp; the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c16">G1</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c17">G2</font> &amp;  <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">comp</a> <font color="Maroon" title="c16">G1</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">comp</a> <font color="Maroon" title="c17">G2</font> &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c16">G1</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="Maroon" title="c17">G2</font> &amp; the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c16">G1</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c17">G2</font> &amp;  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c16">G1</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="Maroon" title="c17">G2</font> )
 
<b>by </b><i><a class="txt" href="ringcat1.html#E10:54"><i><font color="Green" title="E39">A7</font></i></a>, <a class="txt" href="ringcat1.html#E12:54"><i><font color="Green" title="E40">A8</font></i></a>, <a class="ref" href="mcart_1.html#T33" title="MCART_1:th.33">MCART_1:33</a></i>;<br/>
<b>hence </b><a NAME="E15:54"/>
<font color="Maroon" title="c2">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y2</font>
 <b>by </b><i><a class="txt" href="ringcat1.html#E10:54"><i><font color="Green" title="E39">A7</font></i></a>, <a class="txt" href="ringcat1.html#E12:54"><i><font color="Green" title="E40">A8</font></i></a></i>;<br/>


</div>
