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

<b>thus </b><a NAME="E1:68_1_1"/>
for <font color="Olive">a</font> being  <a href="real_1.html#NM1">Real</a><br/> for <font color="Olive">v</font>, <font color="Olive">w</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="hahnban1.html#K10" target="_self">RealVS</a> <font color="Maroon">V</font></span>)</span> holds  <font color="Olive">a</font> <a href="rlvect_1.html#K3">*</a> <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="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">v</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">w</font></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">v</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="hahnban1.html#K10" target="_self">RealVS</a> <font color="Maroon">V</font></span>)</span>, <font color="Maroon">w</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="hahnban1.html#K10" target="_self">RealVS</a> <font color="Maroon">V</font></span>)</span>;<br/>



<a NAME="E1:68_1_1_1"/><i><font color="Green">E35</font></i>: 
 <a href="rlvect_1.html#G1">LoopStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">V</font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">V</font> #) <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#G1">LoopStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="hahnban1.html#K10" target="_self">RealVS</a> <font color="Maroon">V</font></span>)</span>,the <a href="rlvect_1.html#U1">add</a> of <span class="p1">(<span class="default"><a href="hahnban1.html#K10" target="_self">RealVS</a> <font color="Maroon">V</font></span>)</span>,the <a href="struct_0.html#U2">Zero</a> of <span class="p1">(<span class="default"><a href="hahnban1.html#K10" target="_self">RealVS</a> <font color="Maroon">V</font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="hahnban1.html#T4" target="_self">Th4</a></i>;<br/>
<b>set </b><font color="Maroon">a1</font> = <span class="p1"><a href="hahnban1.html#K1" target="_self">[**</a><span class="default"><font color="Maroon">a</font>,0</span><a href="hahnban1.html#K1" target="_self">**]</a></span>;<br/>
<b>reconsider </b><font color="Maroon">v1</font> = <font color="Maroon">v</font>, <font color="Maroon">w1</font> = <font color="Maroon">w</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> <b>by </b><i><a class="ref" href="hahnban1.html#T5" target="_self">Th5</a></i>;<br/>
<b>thus </b><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</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="rlvect_1.html#U2">Mult</a> of <span class="p1">(<span class="default"><a href="hahnban1.html#K10" target="_self">RealVS</a> <font color="Maroon">V</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">w1</font></span>)</span>
<b>by </b><i><a class="ref" href="hahnban1.html#T5" target="_self">Th5</a></i>
<br/>.= 
<span class="p1"><a href="hahnban1.html#K1" target="_self">[**</a><span class="default"><font color="Maroon">a</font>,0</span><a href="hahnban1.html#K1" target="_self">**]</a></span> <a href="vectsp_1.html#K6">*</a> <span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">w1</font></span>)</span>
<b>by </b><i><a class="ref" href="hahnban1.html#T4" target="_self">Th4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="hahnban1.html#K1" target="_self">[**</a><span class="default"><font color="Maroon">a</font>,0</span><a href="hahnban1.html#K1" target="_self">**]</a></span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="hahnban1.html#K1" target="_self">[**</a><span class="default"><font color="Maroon">a</font>,0</span><a href="hahnban1.html#K1" target="_self">**]</a></span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">w1</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D26">VECTSP_1:def 26</a></i>
<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">V</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default">the <a href="rlvect_1.html#U2">Mult</a> of <span class="p3">(<span class="default"><a href="hahnban1.html#K10" target="_self">RealVS</a> <font color="Maroon">V</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">v1</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3"><a href="hahnban1.html#K1" target="_self">[**</a><span class="default"><font color="Maroon">a</font>,0</span><a href="hahnban1.html#K1" target="_self">**]</a></span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">w1</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="hahnban1.html#T4" target="_self">Th4</a></i>
<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <span class="p1">(<span class="default"><a href="hahnban1.html#K10" target="_self">RealVS</a> <font color="Maroon">V</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default">the <a href="rlvect_1.html#U2">Mult</a> of <span class="p3">(<span class="default"><a href="hahnban1.html#K10" target="_self">RealVS</a> <font color="Maroon">V</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">v</font></span><a href="tarski.html#K4">]</a></span></span>)</span>,<span class="p2">(<span class="default">the <a href="rlvect_1.html#U2">Mult</a> of <span class="p3">(<span class="default"><a href="hahnban1.html#K10" target="_self">RealVS</a> <font color="Maroon">V</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">w</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="hahnban1.html#T5" target="_self">Th5</a>, <a class="ref" href="hahnban1.html#T4" target="_self">Th4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font></span>)</span>

;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:68_1_1"/>
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="real_1.html#NM1">Real</a><br/> for <font color="Olive">v</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="hahnban1.html#K10" target="_self">RealVS</a> <font color="Maroon">V</font></span>)</span> holds  <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="real_1.html#K3">+</a> <font color="Olive">b</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">v</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Olive">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">v</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">b</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">v</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="hahnban1.html#K10" target="_self">RealVS</a> <font color="Maroon">V</font></span>)</span>;<br/>



<a NAME="E1:68_1_1_2"/><i><font color="Green">E36</font></i>: 
 <a href="rlvect_1.html#G1">LoopStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">V</font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">V</font> #) <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#G1">LoopStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="hahnban1.html#K10" target="_self">RealVS</a> <font color="Maroon">V</font></span>)</span>,the <a href="rlvect_1.html#U1">add</a> of <span class="p1">(<span class="default"><a href="hahnban1.html#K10" target="_self">RealVS</a> <font color="Maroon">V</font></span>)</span>,the <a href="struct_0.html#U2">Zero</a> of <span class="p1">(<span class="default"><a href="hahnban1.html#K10" target="_self">RealVS</a> <font color="Maroon">V</font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="hahnban1.html#T4" target="_self">Th4</a></i>;<br/>
<b>set </b><font color="Maroon">a1</font> = <span class="p1"><a href="hahnban1.html#K1" target="_self">[**</a><span class="default"><font color="Maroon">a</font>,0</span><a href="hahnban1.html#K1" target="_self">**]</a></span>;<br/>
<b>set </b><font color="Maroon">b1</font> = <span class="p1"><a href="hahnban1.html#K1" target="_self">[**</a><span class="default"><font color="Maroon">b</font>,0</span><a href="hahnban1.html#K1" target="_self">**]</a></span>;<br/>
<a NAME="E2:68_1_1_2"/><i><font color="Green">E37</font></i>: 
<span class="p1"><a href="hahnban1.html#K1" target="_self">[**</a><span class="default"><font color="Maroon">a</font>,0</span><a href="hahnban1.html#K1" target="_self">**]</a></span> <a href="rlvect_1.html#K4">+</a> <span class="p1"><a href="hahnban1.html#K1" target="_self">[**</a><span class="default"><font color="Maroon">b</font>,0</span><a href="hahnban1.html#K1" target="_self">**]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="hahnban1.html#K1" target="_self">[**</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K3">+</a> <font color="Maroon">b</font></span>)</span>,0</span><a href="hahnban1.html#K1" target="_self">**]</a></span>
 
<b>by </b><i><a class="ref" href="complfld.html#T3">COMPLFLD:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">v1</font> = <font color="Maroon">v</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> <b>by </b><i><a class="ref" href="hahnban1.html#T5" target="_self">Th5</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K3">+</a> <font color="Maroon">b</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font> = 
the <a href="rlvect_1.html#U2">Mult</a> of <span class="p1">(<span class="default"><a href="hahnban1.html#K10" target="_self">RealVS</a> <font color="Maroon">V</font></span>)</span> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K3">+</a> <font color="Maroon">b</font></span>)</span>,<font color="Maroon">v</font>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="hahnban1.html#K1" target="_self">[**</a><span class="default"><font color="Maroon">a</font>,0</span><a href="hahnban1.html#K1" target="_self">**]</a></span> <a href="rlvect_1.html#K4">+</a> <span class="p2"><a href="hahnban1.html#K1" target="_self">[**</a><span class="default"><font color="Maroon">b</font>,0</span><a href="hahnban1.html#K1" target="_self">**]</a></span></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v1</font>
<b>by </b><i>, <a class="ref" href="hahnban1.html#T4" target="_self">Th4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="hahnban1.html#K1" target="_self">[**</a><span class="default"><font color="Maroon">a</font>,0</span><a href="hahnban1.html#K1" target="_self">**]</a></span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="hahnban1.html#K1" target="_self">[**</a><span class="default"><font color="Maroon">b</font>,0</span><a href="hahnban1.html#K1" target="_self">**]</a></span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v1</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D26">VECTSP_1:def 26</a></i>
<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <span class="p1">(<span class="default"><a href="hahnban1.html#K10" target="_self">RealVS</a> <font color="Maroon">V</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default">the <a href="rlvect_1.html#U2">Mult</a> of <span class="p3">(<span class="default"><a href="hahnban1.html#K10" target="_self">RealVS</a> <font color="Maroon">V</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">v</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3"><a href="hahnban1.html#K1" target="_self">[**</a><span class="default"><font color="Maroon">b</font>,0</span><a href="hahnban1.html#K1" target="_self">**]</a></span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v1</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="hahnban1.html#T5" target="_self">Th5</a>, <a class="ref" href="hahnban1.html#T4" target="_self">Th4</a></i>
<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <span class="p1">(<span class="default"><a href="hahnban1.html#K10" target="_self">RealVS</a> <font color="Maroon">V</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default">the <a href="rlvect_1.html#U2">Mult</a> of <span class="p3">(<span class="default"><a href="hahnban1.html#K10" target="_self">RealVS</a> <font color="Maroon">V</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">v</font></span><a href="tarski.html#K4">]</a></span></span>)</span>,<span class="p2">(<span class="default">the <a href="rlvect_1.html#U2">Mult</a> of <span class="p3">(<span class="default"><a href="hahnban1.html#K10" target="_self">RealVS</a> <font color="Maroon">V</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">b</font>,<font color="Maroon">v</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="hahnban1.html#T4" target="_self">Th4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span>

;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E3:68_1_1"/>
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="real_1.html#NM1">Real</a><br/> for <font color="Olive">v</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="hahnban1.html#K10" target="_self">RealVS</a> <font color="Maroon">V</font></span>)</span> holds  <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="real_1.html#K4">*</a> <font color="Olive">b</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Olive">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">v</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">b</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">v</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="hahnban1.html#K10" target="_self">RealVS</a> <font color="Maroon">V</font></span>)</span>;<br/>



<a NAME="E1:68_1_1_3"/><i><font color="Green">E38</font></i>: 
 <a href="rlvect_1.html#G1">LoopStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">V</font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">V</font> #) <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#G1">LoopStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="hahnban1.html#K10" target="_self">RealVS</a> <font color="Maroon">V</font></span>)</span>,the <a href="rlvect_1.html#U1">add</a> of <span class="p1">(<span class="default"><a href="hahnban1.html#K10" target="_self">RealVS</a> <font color="Maroon">V</font></span>)</span>,the <a href="struct_0.html#U2">Zero</a> of <span class="p1">(<span class="default"><a href="hahnban1.html#K10" target="_self">RealVS</a> <font color="Maroon">V</font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="hahnban1.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E2:68_1_1_3"/><i><font color="Green">E39</font></i>: 
<span class="p1"><a href="hahnban1.html#K1" target="_self">[**</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font></span>)</span>,0</span><a href="hahnban1.html#K1" target="_self">**]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="hahnban1.html#K1" target="_self">[**</a><span class="default"><font color="Maroon">a</font>,0</span><a href="hahnban1.html#K1" target="_self">**]</a></span> <a href="group_1.html#K10">*</a> <span class="p1"><a href="hahnban1.html#K1" target="_self">[**</a><span class="default"><font color="Maroon">b</font>,0</span><a href="hahnban1.html#K1" target="_self">**]</a></span>
 
<b>by </b><i><a class="ref" href="complfld.html#T6">COMPLFLD:6</a></i>;<br/>
<b>reconsider </b><font color="Maroon">v1</font> = <font color="Maroon">v</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> <b>by </b><i><a class="ref" href="hahnban1.html#T5" target="_self">Th5</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font> = 
the <a href="rlvect_1.html#U2">Mult</a> of <span class="p1">(<span class="default"><a href="hahnban1.html#K10" target="_self">RealVS</a> <font color="Maroon">V</font></span>)</span> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font></span>)</span>,<font color="Maroon">v</font>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="hahnban1.html#K1" target="_self">[**</a><span class="default"><font color="Maroon">a</font>,0</span><a href="hahnban1.html#K1" target="_self">**]</a></span> <a href="group_1.html#K10">*</a> <span class="p2"><a href="hahnban1.html#K1" target="_self">[**</a><span class="default"><font color="Maroon">b</font>,0</span><a href="hahnban1.html#K1" target="_self">**]</a></span></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v1</font>
<b>by </b><i>, <a class="ref" href="hahnban1.html#T4" target="_self">Th4</a></i>
<br/>.= 
<span class="p1"><a href="hahnban1.html#K1" target="_self">[**</a><span class="default"><font color="Maroon">a</font>,0</span><a href="hahnban1.html#K1" target="_self">**]</a></span> <a href="vectsp_1.html#K6">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="hahnban1.html#K1" target="_self">[**</a><span class="default"><font color="Maroon">b</font>,0</span><a href="hahnban1.html#K1" target="_self">**]</a></span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v1</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D26">VECTSP_1:def 26</a></i>
<br/>.= 
the <a href="rlvect_1.html#U2">Mult</a> of <span class="p1">(<span class="default"><a href="hahnban1.html#K10" target="_self">RealVS</a> <font color="Maroon">V</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<span class="p1">(<span class="default"><span class="p2"><a href="hahnban1.html#K1" target="_self">[**</a><span class="default"><font color="Maroon">b</font>,0</span><a href="hahnban1.html#K1" target="_self">**]</a></span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v1</font></span>)</span>
<b>by </b><i><a class="ref" href="hahnban1.html#T4" target="_self">Th4</a></i>
<br/>.= 
the <a href="rlvect_1.html#U2">Mult</a> of <span class="p1">(<span class="default"><a href="hahnban1.html#K10" target="_self">RealVS</a> <font color="Maroon">V</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<span class="p2">(<span class="default">the <a href="rlvect_1.html#U2">Mult</a> of <span class="p3">(<span class="default"><a href="hahnban1.html#K10" target="_self">RealVS</a> <font color="Maroon">V</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">b</font>,<font color="Maroon">v</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="hahnban1.html#T4" target="_self">Th4</a></i>
<br/>.= 
<font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span>

;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">v</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="hahnban1.html#K10" target="_self">RealVS</a> <font color="Maroon">V</font></span>)</span>;<br/>

<a NAME="E4:68_1_1"/>
 <a href="rlvect_1.html#G1">LoopStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">V</font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">V</font> #) <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#G1">LoopStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="hahnban1.html#K10" target="_self">RealVS</a> <font color="Maroon">V</font></span>)</span>,the <a href="rlvect_1.html#U1">add</a> of <span class="p1">(<span class="default"><a href="hahnban1.html#K10" target="_self">RealVS</a> <font color="Maroon">V</font></span>)</span>,the <a href="struct_0.html#U2">Zero</a> of <span class="p1">(<span class="default"><a href="hahnban1.html#K10" target="_self">RealVS</a> <font color="Maroon">V</font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="hahnban1.html#T4" target="_self">Th4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">v1</font> = <font color="Maroon">v</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> ;<br/>
<a NAME="E6:68_1_1"/><i><font color="Green">E40</font></i>: 
<span class="p1"><a href="hahnban1.html#K1" target="_self">[**</a><span class="default">1,0</span><a href="hahnban1.html#K1" target="_self">**]</a></span> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> 
 
<b>by </b><i><a class="ref" href="complfld.html#T10">COMPLFLD:10</a></i>;<br/>
<b>thus </b>1 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font> = 
the <a href="rlvect_1.html#U2">Mult</a> of <span class="p1">(<span class="default"><a href="hahnban1.html#K10" target="_self">RealVS</a> <font color="Maroon">V</font></span>)</span> <a href="binop_1.html#K2">.</a> 1,<font color="Maroon">v</font>

<br/>.= 
<span class="p1"><a href="hahnban1.html#K1" target="_self">[**</a><span class="default">1,0</span><a href="hahnban1.html#K1" target="_self">**]</a></span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v1</font>
<b>by </b><i><a class="ref" href="hahnban1.html#T4" target="_self">Th4</a></i>
<br/>.= 
<font color="Maroon">v</font>
<b>by </b><i><a class="ref" href="hahnban1.html#T5" target="_self">Th5</a>, <a class="ref" href="vectsp_1.html#D26">VECTSP_1:def 26</a></i>
;<br/>


</div>
