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

<b>let </b><font color="Maroon">K</font> be  non <a href="struct_0.html#V3">empty</a> <a href="vectsp_1.html#V7">distributive</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">K</font>, <font color="Maroon">a2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">K</font>, <font color="Maroon">a3</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">K</font>;<br/><b>thus </b>the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a1</font>,<span class="p1">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">K</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a2</font>,<font color="Maroon">a3</font></span>)</span> = 
the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a1</font>,<span class="p1">(<span class="default"><font color="Maroon">a2</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">a3</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T5">RLVECT_1:5</a></i>
<br/>.= 
<font color="Maroon">a1</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">a2</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">a3</font></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D1">GROUP_1:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a2</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a3</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D18">VECTSP_1:def 18</a></i>
<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">K</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a2</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a3</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T5">RLVECT_1:5</a></i>
<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">K</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a1</font>,<font color="Maroon">a2</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a3</font></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D1">GROUP_1:def 1</a></i>
<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">K</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a1</font>,<font color="Maroon">a2</font></span>)</span>,<span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a1</font>,<font color="Maroon">a3</font></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D1">GROUP_1:def 1</a></i>
;<br/><b>thus </b>the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">K</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a1</font>,<font color="Maroon">a2</font></span>)</span>,<font color="Maroon">a3</font> = 
the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">a2</font></span>)</span>,<font color="Maroon">a3</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T5">RLVECT_1:5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">a2</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">a3</font>
<b>by </b><i><a class="ref" href="group_1.html#D1">GROUP_1:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a3</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">a2</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a3</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D18">VECTSP_1:def 18</a></i>
<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">K</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a3</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">a2</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a3</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T5">RLVECT_1:5</a></i>
<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">K</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a1</font>,<font color="Maroon">a3</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">a2</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a3</font></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D1">GROUP_1:def 1</a></i>
<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">K</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a1</font>,<font color="Maroon">a3</font></span>)</span>,<span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a2</font>,<font color="Maroon">a3</font></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D1">GROUP_1:def 1</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:13"/>
the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</font> <a href="binop_1.html#R6">is_distributive_wrt</a> the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">K</font>
 <b>by </b><i><a class="ref" href="binop_1.html#T23">BINOP_1:23</a></i>;<br/>


</div>
