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<b>set </b><font color="Maroon">V</font> =  <a href="vectsp10.html#K1" target="_self">StructVectSp</a> <font color="Maroon">K</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">K</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">K</font>;<br/><b>let </b><font color="Maroon">v</font> be   <a href="vectsp_1.html#NM5">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K1" target="_self">StructVectSp</a> <font color="Maroon">K</font></span>)</span>, <font color="Maroon">w</font> be   <a href="vectsp_1.html#NM5">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K1" target="_self">StructVectSp</a> <font color="Maroon">K</font></span>)</span>;<br/><b>reconsider </b><font color="Maroon">v'</font> = <font color="Maroon">v</font>, <font color="Maroon">w'</font> = <font color="Maroon">w</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">K</font> ;<br/><a NAME="E2:7_1_1_1"/><i><font color="Green">E29</font></i>: 
the <a href="vectsp_1.html#U2">lmult</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K1" target="_self">StructVectSp</a> <font color="Maroon">K</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">w</font>
 <b>by </b><i><a class="ref" href="vectsp_1.html#D24">VECTSP_1:def 24</a></i>;<br/><a NAME="E3:7_1_1_1"/><i><font color="Green">E35</font></i>: 
the <a href="vectsp_1.html#U2">lmult</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K1" target="_self">StructVectSp</a> <font color="Maroon">K</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">y</font>,<font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font>
 <b>by </b><i><a class="ref" href="vectsp_1.html#D24">VECTSP_1:def 24</a></i>;<br/><b>set </b><font color="Maroon">MLT</font> = the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</font>;<br/><b>thus </b><font color="Maroon">x</font> <a href="vectsp_1.html#K6">*</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">w</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">x</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">v'</font>,<font color="Maroon">w'</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D24">VECTSP_1:def 24</a></i>
<br/>.= 
<font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">v'</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">w'</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">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">v'</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">w'</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">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">x</font>,<font color="Maroon">v'</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">w'</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#K1">.</a> <span class="p1">(<span class="default">the <a href="vectsp_1.html#U2">lmult</a> of <span class="p2">(<span class="default"><a href="vectsp10.html#K1" target="_self">StructVectSp</a> <font color="Maroon">K</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<font color="Maroon">v</font></span>)</span>,<span class="p1">(<span class="default">the <a href="vectsp_1.html#U2">lmult</a> of <span class="p2">(<span class="default"><a href="vectsp10.html#K1" target="_self">StructVectSp</a> <font color="Maroon">K</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<font color="Maroon">w</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">x</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">w</font></span>)</span>
<b>by </b><i>, <a class="ref" href="vectsp_1.html#D24">VECTSP_1:def 24</a></i>
;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">y</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font> = 
the <a href="vectsp_1.html#U2">lmult</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K1" target="_self">StructVectSp</a> <font color="Maroon">K</font></span>)</span> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">y</font></span>)</span>,<font color="Maroon">v</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#D24">VECTSP_1:def 24</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">y</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">v'</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">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">v'</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="group_1.html#K1">*</a> <font color="Maroon">v'</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">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">x</font>,<font color="Maroon">v'</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="group_1.html#K1">*</a> <font color="Maroon">v'</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#K1">.</a> <span class="p1">(<span class="default">the <a href="vectsp_1.html#U2">lmult</a> of <span class="p2">(<span class="default"><a href="vectsp10.html#K1" target="_self">StructVectSp</a> <font color="Maroon">K</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<font color="Maroon">v</font></span>)</span>,<span class="p1">(<span class="default">the <a href="vectsp_1.html#U2">lmult</a> of <span class="p2">(<span class="default"><a href="vectsp10.html#K1" target="_self">StructVectSp</a> <font color="Maroon">K</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">y</font>,<font color="Maroon">v</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">x</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font></span>)</span>
<b>by </b><i>, <a class="ref" href="vectsp_1.html#D24">VECTSP_1:def 24</a></i>
;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font> = 
the <a href="vectsp_1.html#U2">lmult</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K1" target="_self">StructVectSp</a> <font color="Maroon">K</font></span>)</span> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font></span>)</span>,<font color="Maroon">v</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#D24">VECTSP_1:def 24</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">v'</font>
<b>by </b><i><a class="ref" href="group_1.html#D1">GROUP_1:def 1</a></i>
<br/>.= 
<font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="group_1.html#K1">*</a> <font color="Maroon">v'</font></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
<br/>.= 
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">x</font>,<span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="group_1.html#K1">*</a> <font color="Maroon">v'</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="vectsp_1.html#U2">lmult</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K1" target="_self">StructVectSp</a> <font color="Maroon">K</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default">the <a href="vectsp_1.html#U2">lmult</a> of <span class="p2">(<span class="default"><a href="vectsp10.html#K1" target="_self">StructVectSp</a> <font color="Maroon">K</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">y</font>,<font color="Maroon">v</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="vectsp_1.html#U2">lmult</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K1" target="_self">StructVectSp</a> <font color="Maroon">K</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D24">VECTSP_1:def 24</a></i>
<br/>.= 
<font color="Maroon">x</font> <a href="vectsp_1.html#K6">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D24">VECTSP_1:def 24</a></i>
;<br/><b>thus </b><span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">K</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</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"><a href="group_1.html#K2">1.</a> <font color="Maroon">K</font></span>)</span>,<font color="Maroon">v'</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#D24">VECTSP_1:def 24</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">K</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">v'</font>
<b>by </b><i><a class="ref" href="group_1.html#D1">GROUP_1:def 1</a></i>
<br/>.= 
<font color="Maroon">v</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#D19">VECTSP_1:def 19</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:7_1_1"/>
 <a href="vectsp10.html#K1" target="_self">StructVectSp</a> <font color="Maroon">K</font> is <a href="vectsp_1.html#V12">VectSp-like</a>
 <b>by </b><i><a class="ref" href="vectsp_1.html#D26">VECTSP_1:def 26</a></i>;<br/>


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