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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="classes2.html#NM1">Universe</a>;<br/>

<b>set </b><font color="Maroon">c<sub>2</sub></font> =  <a href="mod_2.html#K18">Z3</a> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <a href="mod_2.html#K18">Z3</a> , <font color="Maroon">c<sub>4</sub></font> = the <a href="rlvect_1.html#U1">add</a> of <a href="mod_2.html#K18">Z3</a> , <font color="Maroon">c<sub>5</sub></font> =  <a href="vectsp_1.html#K7">comp</a> <a href="mod_2.html#K18">Z3</a> , <font color="Maroon">c<sub>6</sub></font> = the <a href="struct_0.html#U2">Zero</a> of <a href="mod_2.html#K18">Z3</a> , <font color="Maroon">c<sub>7</sub></font> = the <a href="group_1.html#U1">mult</a> of <a href="mod_2.html#K18">Z3</a> , <font color="Maroon">c<sub>8</sub></font> = the <a href="vectsp_1.html#U1">unity</a> of <a href="mod_2.html#K18">Z3</a>  as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="mod_2.html#T40">MOD_2:40</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>9</sub></font> = <span class="p1"><a href="mcart_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="mcart_1.html#K4">]</a></span>;<br/>
<b>set </b><font color="Maroon">c<sub>10</sub></font> = <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><span class="p2"><a href="mcart_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="mcart_1.html#K4">]</a></span>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="mcart_1.html#K3">]</a></span>;<br/>
<a NAME="E2:54"/>
<span class="p1"><a href="mcart_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="mcart_1.html#K4">]</a></span> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="grcat_1.html#T3">GRCAT_1:3</a></i>;<br/>
<a NAME="E3:54"/><b>then </b><i><font color="Green">E34</font></i>: 
<span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><span class="p2"><a href="mcart_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="mcart_1.html#K4">]</a></span>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="mcart_1.html#K3">]</a></span> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="grcat_1.html#T3">GRCAT_1:3</a></i>;<br/>
<a NAME="E4:54"/><i><font color="Green">E35</font></i>: 
 <a href="ringcat1.html#R2" target="_self">GO</a> <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><span class="p2"><a href="mcart_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="mcart_1.html#K4">]</a></span>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="mcart_1.html#K3">]</a></span>, <a href="mod_2.html#K18">Z3</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">c<sub>3</sub></font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="ringcat1.html#D16" target="_self">RINGCAT1:def 16</a><br/>

<b>take </b>
<font color="Maroon">c<sub>4</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>6</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>7</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>8</sub></font>
;<br/>

<b>thus </b><a NAME="E1:54_1"/>
( <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><span class="p2"><a href="mcart_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="mcart_1.html#K4">]</a></span>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="mcart_1.html#K3">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><span class="p2"><a href="mcart_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="mcart_1.html#K4">]</a></span>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="mcart_1.html#K3">]</a></span> &amp; ex <font color="Olive">b<sub>1</sub></font> being <a href="vectsp_1.html#V3">strict</a> <a href="funcsdom.html#NM1">Ring</a> st <br/>(  <a href="mod_2.html#K18">Z3</a>  <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> the <a href="rlvect_1.html#U1">add</a> of <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="vectsp_1.html#K7">comp</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U2">Zero</a> of <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> the <a href="group_1.html#U1">mult</a> of <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> the <a href="vectsp_1.html#U1">unity</a> of <font color="Olive">b<sub>1</sub></font> ) )
 ;<br/>


</div><b>end;</b></div>
<b>take </b>
<span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><span class="p2"><a href="mcart_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="mcart_1.html#K4">]</a></span>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="mcart_1.html#K3">]</a></span>
;<br/>

<b>thus </b><a NAME="E5:54"/>
( <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><span class="p2"><a href="mcart_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="mcart_1.html#K4">]</a></span>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="mcart_1.html#K3">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="ringcat1.html#R2" target="_self">GO</a> <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><span class="p2"><a href="mcart_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="mcart_1.html#K4">]</a></span>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="mcart_1.html#K3">]</a></span>, <a href="mod_2.html#K18">Z3</a>  )
 <b>by </b><i><a class="txt" href="ringcat1.html#E3:54"><i><font color="Green">E34</font></i></a>, <a class="txt" href="ringcat1.html#E4:54"><i><font color="Green">E35</font></i></a></i>;<br/>


</div>
