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<b>let </b><font color="Maroon">I</font> be  non <a href="struct_0.html#V3">empty</a> <a href="rlvect_1.html#V3">Abelian</a> <a href="group_1.html#V4">associative</a> <a href="group_1.html#V7">commutative</a> non <a href="vectsp_1.html#V10">degenerated</a> <a href="vectsp_2.html#V2">domRing-like</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon">u</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="quofield.html#K1" target="_self">Q.</a> <font color="Maroon">I</font>, <font color="Maroon">v</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="quofield.html#K1" target="_self">Q.</a> <font color="Maroon">I</font>, <font color="Maroon">w</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="quofield.html#K1" target="_self">Q.</a> <font color="Maroon">I</font>;<br/>



<a NAME="E1:9"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">u</font> <a href="quofield.html#K3" target="_self">`2</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">I</font> &amp; <font color="Maroon">v</font> <a href="quofield.html#K3" target="_self">`2</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">I</font> &amp; <font color="Maroon">w</font> <a href="quofield.html#K3" target="_self">`2</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">I</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E2:9"/><b>then </b><i><font color="Green">E23</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">I</font>
 
<b>by </b><i><a class="ref" href="vectsp_2.html#D5">VECTSP_2:def 5</a></i>;<br/>
<a NAME="E3:9"/><i><font color="Green">E24</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">I</font>
 
<b>by </b><i>, <a class="ref" href="vectsp_2.html#D5">VECTSP_2:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">t</font> = <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">v</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><font color="Maroon">w</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">v</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><font color="Maroon">w</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span> as    <a href="subset_1.html#M2">Element</a> of  <a href="quofield.html#K1" target="_self">Q.</a> <font color="Maroon">I</font> <b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">s</font> = <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">u</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><font color="Maroon">v</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">u</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><font color="Maroon">v</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span> as    <a href="subset_1.html#M2">Element</a> of  <a href="quofield.html#K1" target="_self">Q.</a> <font color="Maroon">I</font> <b>by </b><i>, </i>;<br/>
<i><font color="Green">E25</font></i>:  <a href="quofield.html#K5" target="_self">pmult</a> <font color="Maroon">u</font>,<span class="p1">(<span class="default"><a href="quofield.html#K5" target="_self">pmult</a> <font color="Maroon">v</font>,<font color="Maroon">w</font></span>)</span> = 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">u</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">v</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">u</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">u</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">v</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">u</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">v</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">u</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p4">(<span class="default"><font color="Maroon">v</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><font color="Maroon">w</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">u</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">v</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">u</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p4">(<span class="default"><font color="Maroon">v</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><font color="Maroon">w</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">u</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p4">(<span class="default"><font color="Maroon">v</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><font color="Maroon">w</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><font color="Maroon">w</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">u</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p4">(<span class="default"><font color="Maroon">v</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><font color="Maroon">w</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
<br/>.= 
 <a href="quofield.html#K5" target="_self">pmult</a> <span class="p1">(<span class="default"><a href="quofield.html#K5" target="_self">pmult</a> <font color="Maroon">u</font>,<font color="Maroon">v</font></span>)</span>,<font color="Maroon">w</font>
<b>by </b><i><a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>
;<br/>
 <a href="quofield.html#K5" target="_self">pmult</a> <font color="Maroon">u</font>,<font color="Maroon">v</font> = 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">u</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><font color="Maroon">v</font> <a href="quofield.html#K2" target="_self">`1</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">u</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><font color="Maroon">v</font> <a href="quofield.html#K3" target="_self">`2</a> </span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span>

<br/>.= 
 <a href="quofield.html#K5" target="_self">pmult</a> <font color="Maroon">v</font>,<font color="Maroon">u</font>

;<br/>
<b>hence </b><a NAME="E8:9"/>
(  <a href="quofield.html#K5" target="_self">pmult</a> <font color="Maroon">u</font>,<span class="p1">(<span class="default"><a href="quofield.html#K5" target="_self">pmult</a> <font color="Maroon">v</font>,<font color="Maroon">w</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="quofield.html#K5" target="_self">pmult</a> <span class="p1">(<span class="default"><a href="quofield.html#K5" target="_self">pmult</a> <font color="Maroon">u</font>,<font color="Maroon">v</font></span>)</span>,<font color="Maroon">w</font> &amp;  <a href="quofield.html#K5" target="_self">pmult</a> <font color="Maroon">u</font>,<font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="quofield.html#K5" target="_self">pmult</a> <font color="Maroon">v</font>,<font color="Maroon">u</font> )
 <b>by </b><i/>;<br/>


</div>
