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

<b>let </b><font color="Maroon" title="c1">l</font> be    <a href="bhsp_1.html#L1" title="BHSP_1:struct.1">UNITSTR</a> ;<br/>

<b>assume </b><a NAME="E1:31"/><i><font color="Green" title="E28">A1</font></i>: 
 <a href="rlvect_1.html#G2" title="RLVECT_1:aggr.2">RLSStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">l</font>,<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">l</font></span>)</span>,the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c1">l</font>,the <a href="rlvect_1.html#U2" title="RLVECT_1:sel.2">Mult</a> of <font color="Maroon" title="c1">l</font> #) is   <a href="rlvect_1.html#NM2" title="RLVECT_1:NM.2">RealLinearSpace</a>
 ;<br/>

<a NAME="E2:31"/>
not the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">l</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 
<b>by </b><i><a class="txt" href="rsspace.html#E1:31"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="struct_0.html#D1" title="STRUCT_0:def.1">STRUCT_0:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c2">l</font> = <font color="Maroon" title="c1">l</font> as  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="rlvect_1.html#L2" title="RLVECT_1:struct.2">RLSStruct</a>  <b>by </b><i><a class="ref" href="struct_0.html#D1" title="STRUCT_0:def.1">STRUCT_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c3">l0</font> =  <a href="rlvect_1.html#G2" title="RLVECT_1:aggr.2">RLSStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">l</font>,<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">l</font></span>)</span>,the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c2">l</font>,the <a href="rlvect_1.html#U2" title="RLVECT_1:sel.2">Mult</a> of <font color="Maroon" title="c2">l</font> #) as   <a href="rlvect_1.html#NM2" title="RLVECT_1:NM.2">RealLinearSpace</a> <b>by </b><i><a class="txt" href="rsspace.html#E1:31"><i><font color="Green" title="E28">A1</font></i></a></i>;<br/>
<a NAME="E5:31"/><i><font color="Green" title="E29">A2</font></i>: 
for <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c2">l</font> holds  <font color="Olive" title="b1">v</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Olive" title="b2">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">w</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Olive" title="b1">v</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">v</font>, <font color="Maroon" title="c5">w</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c2">l</font>;<br/>

<b>reconsider </b><font color="Maroon" title="c6">v1</font> = <font color="Maroon" title="c4">v</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c3">l0</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c7">w1</font> = <font color="Maroon" title="c5">w</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c3">l0</font> ;<br/>
<b>thus </b><font color="Maroon" title="c4">v</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c5">w</font> = 
<font color="Maroon" title="c6">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">w1</font>

<br/>.= 
<font color="Maroon" title="c5">w</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c4">v</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D3" title="RLVECT_1:def.3">RLVECT_1:def 3</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E6:31"/><i><font color="Green" title="E30">A3</font></i>: 
for <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">v</font>, <font color="Olive" title="b3">w</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c2">l</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">u</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Olive" title="b2">v</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Olive" title="b3">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">u</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">v</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Olive" title="b3">w</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">u</font>, <font color="Maroon" title="c5">v</font>, <font color="Maroon" title="c6">w</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c2">l</font>;<br/>

<b>reconsider </b><font color="Maroon" title="c7">u1</font> = <font color="Maroon" title="c4">u</font>, <font color="Maroon" title="c8">v1</font> = <font color="Maroon" title="c5">v</font>, <font color="Maroon" title="c9">w1</font> = <font color="Maroon" title="c6">w</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c3">l0</font> ;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c4">u</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c5">v</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c6">w</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">u1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c8">v1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c9">w1</font>

<br/>.= 
<font color="Maroon" title="c7">u1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c9">w1</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></i>
<br/>.= 
<font color="Maroon" title="c4">u</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c6">w</font></span>)</span>

;<br/>


</div><b>end;</b></div>
<a NAME="E7:31"/><i><font color="Green" title="E31">A4</font></i>: 
for <font color="Olive" title="b1">v</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c2">l</font> holds  <font color="Olive" title="b1">v</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">l</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">v</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c2">l</font>;<br/>

<b>reconsider </b><font color="Maroon" title="c5">v1</font> = <font color="Maroon" title="c4">v</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c3">l0</font> ;<br/>
<b>thus </b><font color="Maroon" title="c4">v</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">l</font></span>)</span> = 
<font color="Maroon" title="c5">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">l0</font></span>)</span>

<br/>.= 
<font color="Maroon" title="c4">v</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D7" title="RLVECT_1:def.7">RLVECT_1:def 7</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E8:31"/><i><font color="Green" title="E32">A5</font></i>: 
for <font color="Olive" title="b1">v</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c2">l</font>  ex <font color="Olive" title="b2">w</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c2">l</font> st <font color="Olive" title="b1">v</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Olive" title="b2">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">l</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">v</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c2">l</font>;<br/>

<b>reconsider </b><font color="Maroon" title="c5">v1</font> = <font color="Maroon" title="c4">v</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c3">l0</font> ;<br/>
<b>consider </b><font color="Maroon" title="c6">w1</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c3">l0</font><b> such that </b><br/><a NAME="E3:31_4"/><i><font color="Green" title="E33">A6</font></i>: 
<font color="Maroon" title="c5">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c6">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">l0</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D8" title="RLVECT_1:def.8">RLVECT_1:def 8</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">w</font> = <font color="Maroon" title="c6">w1</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c2">l</font> ;<br/>
<b>take </b>
<font color="Maroon" title="c7">w</font>
;<br/>

<b>thus </b><a NAME="E5:31_4"/>
<font color="Maroon" title="c4">v</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c7">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">l</font>
 <b>by </b><i><a class="txt" href="rsspace.html#E3:31_4"><i><font color="Green" title="E33">A6</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:31"/><i><font color="Green" title="E33">A7</font></i>: 
for <font color="Olive" title="b1">a</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><br/> for <font color="Olive" title="b2">v</font>, <font color="Olive" title="b3">w</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c2">l</font> holds  <font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">v</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Olive" title="b3">w</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Olive" title="b2">v</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Olive" title="b3">w</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">a</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>let </b><font color="Maroon" title="c5">v</font>, <font color="Maroon" title="c6">w</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c2">l</font>;<br/>



<b>reconsider </b><font color="Maroon" title="c7">v1</font> = <font color="Maroon" title="c5">v</font>, <font color="Maroon" title="c8">w1</font> = <font color="Maroon" title="c6">w</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c3">l0</font> ;<br/>
<b>thus </b><font color="Maroon" title="c4">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c6">w</font></span>)</span> = 
<font color="Maroon" title="c4">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c8">w1</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c7">v1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c8">w1</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9" title="RLVECT_1:def.9">RLVECT_1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c5">v</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c6">w</font></span>)</span>

;<br/>


</div><b>end;</b></div>
<a NAME="E10:31"/><i><font color="Green" title="E34">A8</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><br/> for <font color="Olive" title="b3">v</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c2">l</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Olive" title="b3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Olive" title="b3">v</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">b</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Olive" title="b3">v</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>let </b><font color="Maroon" title="c6">v</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c2">l</font>;<br/>



<b>reconsider </b><font color="Maroon" title="c7">v1</font> = <font color="Maroon" title="c6">v</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c3">l0</font> ;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c6">v</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c7">v1</font>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c7">v1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c7">v1</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9" title="RLVECT_1:def.9">RLVECT_1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c6">v</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c6">v</font></span>)</span>

;<br/>


</div><b>end;</b></div>
<a NAME="E11:31"/><i><font color="Green" title="E35">A9</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><br/> for <font color="Olive" title="b3">v</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c2">l</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Olive" title="b3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">b</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Olive" title="b3">v</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>let </b><font color="Maroon" title="c6">v</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c2">l</font>;<br/>



<b>reconsider </b><font color="Maroon" title="c7">v1</font> = <font color="Maroon" title="c6">v</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c3">l0</font> ;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c6">v</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c7">v1</font>

<br/>.= 
<font color="Maroon" title="c4">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c7">v1</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9" title="RLVECT_1:def.9">RLVECT_1:def 9</a></i>
<br/>.= 
<font color="Maroon" title="c4">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c6">v</font></span>)</span>

;<br/>


</div><b>end;</b></div>
<a NAME="E12:31"/>
for <font color="Olive" title="b1">v</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c2">l</font> holds  1 <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">v</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c2">l</font>;<br/>

<b>reconsider </b><font color="Maroon" title="c5">v1</font> = <font color="Maroon" title="c4">v</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c3">l0</font> ;<br/>
<b>thus </b>1 <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c4">v</font> = 
1 <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c5">v1</font>

<br/>.= 
<font color="Maroon" title="c4">v</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9" title="RLVECT_1:def.9">RLVECT_1:def 9</a></i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E13:31"/>
<font color="Maroon" title="c1">l</font> is   <a href="rlvect_1.html#NM2" title="RLVECT_1:NM.2">RealLinearSpace</a>
 <b>by </b><i><a class="txt" href="rsspace.html#E5:31"><i><font color="Green" title="E29">A2</font></i></a>, <a class="txt" href="rsspace.html#E6:31"><i><font color="Green" title="E30">A3</font></i></a>, <a class="txt" href="rsspace.html#E7:31"><i><font color="Green" title="E31">A4</font></i></a>, <a class="txt" href="rsspace.html#E8:31"><i><font color="Green" title="E32">A5</font></i></a>, <a class="txt" href="rsspace.html#E9:31"><i><font color="Green" title="E33">A7</font></i></a>, <a class="txt" href="rsspace.html#E10:31"><i><font color="Green" title="E34">A8</font></i></a>, <a class="txt" href="rsspace.html#E11:31"><i><font color="Green" title="E35">A9</font></i></a>, <a class="ref" href="rlvect_1.html#T7" title="RLVECT_1:th.7">RLVECT_1:7</a></i>;<br/>


</div>
