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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:2_2_1"/><i><font color="Green">E1</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="group_1.html#V7">commutative</a>
 <b>and </b><br/><a NAME="E2:2_2_1"/><i><font color="Green">E2</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="vectsp_1.html#V5">left-distributive</a>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="vectsp_1.html#D18">VECTSP_1:def 18</a><br/>

<b>thus </b><font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="gcd_1.html#E1:2_2_1"><i><font color="Green">E1</font></i></a>, <a class="ref" href="group_1.html#D16">GROUP_1:def 16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="gcd_1.html#E2:2_2_1"><i><font color="Green">E2</font></i></a>, <a class="ref" href="vectsp_1.html#D12">VECTSP_1:def 12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="gcd_1.html#E1:2_2_1"><i><font color="Green">E1</font></i></a>, <a class="ref" href="group_1.html#D16">GROUP_1:def 16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="gcd_1.html#E1:2_2_1"><i><font color="Green">E1</font></i></a>, <a class="ref" href="group_1.html#D16">GROUP_1:def 16</a></i>
;<br/>

<b>thus </b><a NAME="E4:2_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="gcd_1.html#E2:2_2_1"><i><font color="Green">E2</font></i></a>, <a class="ref" href="vectsp_1.html#D12">VECTSP_1:def 12</a></i>;<br/>


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