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

<b>let </b><font color="Maroon">R</font> be   <a href="funcsdom.html#NM1">Ring</a>;<br/>

<b>set </b><font color="Maroon">GF</font> =  <a href="rlvect_1.html#G1">LoopStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">R</font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">R</font> #);<br/>
<a NAME="E1:66"/>
 <a href="rlvect_1.html#G1">LoopStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">R</font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">R</font> #) <a href="hidden.html#R1">=</a>  <a href="vectsp_2.html#K3" target="_self">AbGr</a> <font color="Maroon">R</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">GF</font> =  <a href="rlvect_1.html#G1">LoopStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">R</font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">R</font> #) as   <a href="vectsp_1.html#NM1">AbGroup</a> ;<br/>
<b>set </b><font color="Maroon">MLT</font> = the <a href="group_1.html#U1">mult</a> of <font color="Maroon">R</font>;<br/>
<b>set </b><font color="Maroon">LS</font> =  <a href="vectsp_2.html#G1" target="_self">RightModStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">R</font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">R</font>,the <a href="group_1.html#U1">mult</a> of <font color="Maroon">R</font> #);<br/>
<a NAME="E3:66"/>
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="vectsp_1.html#NM3">Scalar</a> of <font color="Maroon">R</font><br/> for <font color="Olive">v</font>, <font color="Olive">w</font> being  <a href="vectsp_2.html#NM5">Vector</a> of <a href="vectsp_2.html#G1" target="_self">RightModStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">R</font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">R</font>,the <a href="group_1.html#U1">mult</a> of <font color="Maroon">R</font> #) holds <br/> ( <span class="p1">(<span class="default"><font color="Olive">v</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">w</font></span>)</span> <a href="vectsp_2.html#K6" target="_self">*</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">v</font> <a href="vectsp_2.html#K6" target="_self">*</a> <font color="Olive">x</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Olive">w</font> <a href="vectsp_2.html#K6" target="_self">*</a> <font color="Olive">x</font></span>)</span> &amp; <font color="Olive">v</font> <a href="vectsp_2.html#K6" target="_self">*</a> <span class="p1">(<span class="default"><font color="Olive">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">y</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">v</font> <a href="vectsp_2.html#K6" target="_self">*</a> <font color="Olive">x</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Olive">v</font> <a href="vectsp_2.html#K6" target="_self">*</a> <font color="Olive">y</font></span>)</span> &amp; <font color="Olive">v</font> <a href="vectsp_2.html#K6" target="_self">*</a> <span class="p1">(<span class="default"><font color="Olive">y</font> <a href="group_1.html#K1">*</a> <font color="Olive">x</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">v</font> <a href="vectsp_2.html#K6" target="_self">*</a> <font color="Olive">y</font></span>)</span> <a href="vectsp_2.html#K6" target="_self">*</a> <font color="Olive">x</font> &amp; <font color="Olive">v</font> <a href="vectsp_2.html#K6" target="_self">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">R</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">v</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="vectsp_1.html#NM3">Scalar</a> of <font color="Maroon">R</font>, <font color="Maroon">y</font> be   <a href="vectsp_1.html#NM3">Scalar</a> of <font color="Maroon">R</font>;<br/><b>let </b><font color="Maroon">v</font> be   <a href="vectsp_2.html#NM5">Vector</a> of <a href="vectsp_2.html#G1" target="_self">RightModStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">R</font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">R</font>,the <a href="group_1.html#U1">mult</a> of <font color="Maroon">R</font> #), <font color="Maroon">w</font> be   <a href="vectsp_2.html#NM5">Vector</a> of <a href="vectsp_2.html#G1" target="_self">RightModStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">R</font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">R</font>,the <a href="group_1.html#U1">mult</a> of <font color="Maroon">R</font> #);<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="vectsp_1.html#NM3">Scalar</a> of <font color="Maroon">R</font> ;<br/>
<b>thus </b><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> <a href="vectsp_2.html#K6" target="_self">*</a> <font color="Maroon">x</font> = 
the <a href="group_1.html#U1">mult</a> of <font color="Maroon">R</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">R</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">v'</font>,<font color="Maroon">w'</font></span>)</span>,<font color="Maroon">x</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">v'</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">w'</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">x</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">v'</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">w'</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x</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">GF</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="vectsp_2.html#K6" target="_self">*</a> <font color="Maroon">x</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="vectsp_2.html#K6" target="_self">*</a> <font color="Maroon">x</font></span>)</span>
<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">v</font> <a href="vectsp_2.html#K6" target="_self">*</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="vectsp_2.html#K6" target="_self">*</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T5">RLVECT_1:5</a></i>
;<br/>

<b>thus </b><font color="Maroon">v</font> <a href="vectsp_2.html#K6" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font></span>)</span> = 
<font color="Maroon">v'</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">v'</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">v'</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y</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">GF</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="vectsp_2.html#K6" target="_self">*</a> <font color="Maroon">x</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="vectsp_2.html#K6" target="_self">*</a> <font color="Maroon">y</font></span>)</span>
<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">v</font> <a href="vectsp_2.html#K6" target="_self">*</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="vectsp_2.html#K6" target="_self">*</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T5">RLVECT_1:5</a></i>
;<br/>

<b>thus </b><font color="Maroon">v</font> <a href="vectsp_2.html#K6" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font></span>)</span> = 
<font color="Maroon">v'</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">x</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">v'</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">x</font>
<b>by </b><i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="vectsp_2.html#K6" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="vectsp_2.html#K6" target="_self">*</a> <font color="Maroon">x</font>

;<br/>

<b>thus </b><font color="Maroon">v</font> <a href="vectsp_2.html#K6" target="_self">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">R</font></span>)</span> = 
<font color="Maroon">v'</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">R</font></span>)</span>

<br/>.= 
<font color="Maroon">v</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#D13">VECTSP_1:def 13</a></i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:66"/>
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="vectsp_1.html#NM3">Scalar</a> of <font color="Maroon">R</font><br/> for <font color="Olive">v</font>, <font color="Olive">w</font> being  <a href="vectsp_2.html#NM5">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp_2.html#K5" target="_self">RightModule</a> <font color="Maroon">R</font></span>)</span> holds <br/> ( <span class="p1">(<span class="default"><font color="Olive">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">w</font></span>)</span> <a href="vectsp_2.html#K6" target="_self">*</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">v</font> <a href="vectsp_2.html#K6" target="_self">*</a> <font color="Olive">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Olive">w</font> <a href="vectsp_2.html#K6" target="_self">*</a> <font color="Olive">x</font></span>)</span> &amp; <font color="Olive">v</font> <a href="vectsp_2.html#K6" target="_self">*</a> <span class="p1">(<span class="default"><font color="Olive">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">y</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">v</font> <a href="vectsp_2.html#K6" target="_self">*</a> <font color="Olive">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Olive">v</font> <a href="vectsp_2.html#K6" target="_self">*</a> <font color="Olive">y</font></span>)</span> &amp; <font color="Olive">v</font> <a href="vectsp_2.html#K6" target="_self">*</a> <span class="p1">(<span class="default"><font color="Olive">y</font> <a href="group_1.html#K1">*</a> <font color="Olive">x</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">v</font> <a href="vectsp_2.html#K6" target="_self">*</a> <font color="Olive">y</font></span>)</span> <a href="vectsp_2.html#K6" target="_self">*</a> <font color="Olive">x</font> &amp; <font color="Olive">v</font> <a href="vectsp_2.html#K6" target="_self">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">R</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">v</font> )
 ;<br/>


</div>
