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<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y2</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:53"/><i><font color="Green">E31</font></i>: 
 <a href="ringcat1.html#R2" target="_self">GO</a> <font color="Maroon">x</font>,<font color="Maroon">y1</font>
 <b>and </b><br/><a NAME="E2:53"/><i><font color="Green">E32</font></i>: 
 <a href="ringcat1.html#R2" target="_self">GO</a> <font color="Maroon">x</font>,<font color="Maroon">y2</font>
 ;<br/>

<b>consider </b><font color="Maroon">a1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">a2</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">a3</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">a4</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">a5</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">a6</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:53"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">x</font> <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">a1</font>,<font color="Maroon">a2</font>,<font color="Maroon">a3</font>,<font color="Maroon">a4</font></span><a href="mcart_1.html#K4">]</a></span>,<font color="Maroon">a5</font>,<font color="Maroon">a6</font></span><a href="mcart_1.html#K3">]</a></span>
 <b>and </b><br/><a NAME="E5:53"/><i><font color="Green">E35</font></i>: 
 ex <font color="Olive">G</font> being <a href="vectsp_1.html#V3">strict</a> <a href="funcsdom.html#NM1">Ring</a> st <br/>( <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Olive">G</font> &amp; <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">G</font> &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> the <a href="rlvect_1.html#U1">add</a> of <font color="Olive">G</font> &amp; <font color="Maroon">a3</font> <a href="hidden.html#R1">=</a>  <a href="vectsp_1.html#K7">comp</a> <font color="Olive">G</font> &amp; <font color="Maroon">a4</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U2">Zero</a> of <font color="Olive">G</font> &amp; <font color="Maroon">a5</font> <a href="hidden.html#R1">=</a> the <a href="group_1.html#U1">mult</a> of <font color="Olive">G</font> &amp; <font color="Maroon">a6</font> <a href="hidden.html#R1">=</a> the <a href="vectsp_1.html#U1">unity</a> of <font color="Olive">G</font> )
 <b>by </b><i>, <a class="txt" href="ringcat1.html#E1:53"><i><font color="Green">E31</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">b1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">b2</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">b3</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">b4</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">b5</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">b6</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:53"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">x</font> <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">b1</font>,<font color="Maroon">b2</font>,<font color="Maroon">b3</font>,<font color="Maroon">b4</font></span><a href="mcart_1.html#K4">]</a></span>,<font color="Maroon">b5</font>,<font color="Maroon">b6</font></span><a href="mcart_1.html#K3">]</a></span>
 <b>and </b><br/><a NAME="E8:53"/><i><font color="Green">E37</font></i>: 
 ex <font color="Olive">G</font> being <a href="vectsp_1.html#V3">strict</a> <a href="funcsdom.html#NM1">Ring</a> st <br/>( <font color="Maroon">y2</font> <a href="hidden.html#R1">=</a> <font color="Olive">G</font> &amp; <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">G</font> &amp; <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a> the <a href="rlvect_1.html#U1">add</a> of <font color="Olive">G</font> &amp; <font color="Maroon">b3</font> <a href="hidden.html#R1">=</a>  <a href="vectsp_1.html#K7">comp</a> <font color="Olive">G</font> &amp; <font color="Maroon">b4</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U2">Zero</a> of <font color="Olive">G</font> &amp; <font color="Maroon">b5</font> <a href="hidden.html#R1">=</a> the <a href="group_1.html#U1">mult</a> of <font color="Olive">G</font> &amp; <font color="Maroon">b6</font> <a href="hidden.html#R1">=</a> the <a href="vectsp_1.html#U1">unity</a> of <font color="Olive">G</font> )
 <b>by </b><i><a class="txt" href="ringcat1.html#E4:53"><i><font color="Green">E34</font></i></a>, <a class="txt" href="ringcat1.html#E1:53"><i><font color="Green">E31</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">G1</font> being  <a href="vectsp_1.html#V3">strict</a> <a href="funcsdom.html#NM1">Ring</a><b> such that </b><br/><a NAME="E10:53"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> &amp; <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G1</font> &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">G1</font> &amp; <font color="Maroon">a3</font> <a href="hidden.html#R1">=</a>  <a href="vectsp_1.html#K7">comp</a> <font color="Maroon">G1</font> &amp; <font color="Maroon">a4</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">G1</font> &amp; <font color="Maroon">a5</font> <a href="hidden.html#R1">=</a> the <a href="group_1.html#U1">mult</a> of <font color="Maroon">G1</font> &amp; <font color="Maroon">a6</font> <a href="hidden.html#R1">=</a> the <a href="vectsp_1.html#U1">unity</a> of <font color="Maroon">G1</font> )
 <b>by </b><i><a class="txt" href="ringcat1.html#E7:53"><i><font color="Green">E36</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">G2</font> being  <a href="vectsp_1.html#V3">strict</a> <a href="funcsdom.html#NM1">Ring</a><b> such that </b><br/><a NAME="E12:53"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">y2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G2</font> &amp; <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G2</font> &amp; <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a> the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">G2</font> &amp; <font color="Maroon">b3</font> <a href="hidden.html#R1">=</a>  <a href="vectsp_1.html#K7">comp</a> <font color="Maroon">G2</font> &amp; <font color="Maroon">b4</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">G2</font> &amp; <font color="Maroon">b5</font> <a href="hidden.html#R1">=</a> the <a href="group_1.html#U1">mult</a> of <font color="Maroon">G2</font> &amp; <font color="Maroon">b6</font> <a href="hidden.html#R1">=</a> the <a href="vectsp_1.html#U1">unity</a> of <font color="Maroon">G2</font> )
 <b>by </b><i/>;<br/>
<a NAME="E13:53"/>
( <span class="p1"><a href="mcart_1.html#K4">[</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">a2</font>,<font color="Maroon">a3</font>,<font color="Maroon">a4</font></span><a href="mcart_1.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="mcart_1.html#K4">[</a><span class="default"><font color="Maroon">b1</font>,<font color="Maroon">b2</font>,<font color="Maroon">b3</font>,<font color="Maroon">b4</font></span><a href="mcart_1.html#K4">]</a></span> &amp; <font color="Maroon">a5</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b5</font> &amp; <font color="Maroon">a6</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b6</font> )
 
<b>by </b><i><a class="txt" href="ringcat1.html#E5:53"><i><font color="Green">E35</font></i></a>, <a class="txt" href="ringcat1.html#E8:53"><i><font color="Green">E37</font></i></a>, <a class="ref" href="mcart_1.html#T28">MCART_1:28</a></i>;<br/>
<a NAME="E14:53"/><b>then </b>
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G1</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G2</font> &amp; the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">G1</font> <a href="funct_2.html#R1">=</a> the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">G2</font> &amp;  <a href="vectsp_1.html#K7">comp</a> <font color="Maroon">G1</font> <a href="funct_2.html#R1">=</a>  <a href="vectsp_1.html#K7">comp</a> <font color="Maroon">G2</font> &amp; the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">G1</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">G2</font> &amp; the <a href="group_1.html#U1">mult</a> of <font color="Maroon">G1</font> <a href="funct_2.html#R1">=</a> the <a href="group_1.html#U1">mult</a> of <font color="Maroon">G2</font> &amp; the <a href="vectsp_1.html#U1">unity</a> of <font color="Maroon">G1</font> <a href="hidden.html#R1">=</a> the <a href="vectsp_1.html#U1">unity</a> of <font color="Maroon">G2</font> )
 
<b>by </b><i><a class="txt" href="ringcat1.html#E9"><i><font color="Green">Lemma39</font></i></a>, <a class="ref" href="ringcat1.html#D6" target="_self">Def6</a>, <a class="ref" href="mcart_1.html#T33">MCART_1:33</a></i>;<br/>
<b>hence </b><a NAME="E15:53"/>
<font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y2</font>
 <b>by </b><i><a class="txt" href="ringcat1.html#E9"><i><font color="Green">Lemma39</font></i></a>, <a class="ref" href="ringcat1.html#D6" target="_self">Def6</a></i>;<br/>


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