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

<b>take </b>
<font color="Maroon" title="c3">X</font>
;<br/>

<b>thus </b><a NAME="E1:125_1_1"/>
<font color="Maroon" title="c3">X</font> is <a href="clvect_1.html#V5" title="CLVECT_1:attr.5">ComplexNormSpace-like</a>
 <b>by </b><i><a class="ref" href="clvect_1.html#D10" target="_self" title="CLVECT_1:def.10">Def10</a>, <a class="txt" href="clvect_1.html#E127"><i><font color="Green" title="E73">Lm12</font></i></a></i>;<br/>

<a NAME="E2:125_1_1"/><i><font color="Green" title="E75">A1</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c3">X</font><br/> for <font color="Olive" title="b3">x'</font>, <font color="Olive" title="b4">y'</font> being  <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="clvect_1.html#K2" title="CLVECT_1:func.2">(0).</a> <font color="Maroon" title="c1">V</font></span>)</span>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">x'</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">y'</font> holds <br/>( <font color="Olive" title="b1">x</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">x'</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Olive" title="b4">y'</font> &amp; ( for <font color="Olive" title="b5">z</font> being  <a href="cfunct_1.html#NM1" title="CFUNCT_1:NM.1">Complex</a> holds  <font color="Olive" title="b5">z</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">z</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Olive" title="b3">x'</font> ) )
 
;<br/>
<b>thus </b><a NAME="E3:125_1_1"/>
<font color="Maroon" title="c3">X</font> is <a href="clvect_1.html#V2" title="CLVECT_1:attr.2">ComplexLinearSpace-like</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="125_1_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:125_1_1_1"/>
for <font color="Olive" title="b1">z</font> being  <a href="cfunct_1.html#NM1" title="CFUNCT_1:NM.1">Complex</a><br/> for <font color="Olive" title="b2">v</font>, <font color="Olive" title="b3">w</font> being  <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c3">X</font> holds  <font color="Olive" title="b1">z</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">v</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</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">z</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Olive" title="b2">v</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">z</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Olive" title="b3">w</font></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="clvect_1.html#D2" target="_self" title="CLVECT_1:def.2">CLVECT_1:def 2</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="125_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">z</font> be   <a href="cfunct_1.html#NM1" title="CFUNCT_1:NM.1">Complex</a>;<br/><b>let </b><font color="Maroon" title="c5">v</font>, <font color="Maroon" title="c6">w</font> be   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c3">X</font>;<br/>



<b>reconsider </b><font color="Maroon" title="c7">v'</font> = <font color="Maroon" title="c5">v</font>, <font color="Maroon" title="c8">w'</font> = <font color="Maroon" title="c6">w</font> as   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="clvect_1.html#K2" title="CLVECT_1:func.2">(0).</a> <font color="Maroon" title="c1">V</font></span>)</span> ;<br/>
<b>thus </b><font color="Maroon" title="c4">z</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <font color="Maroon" title="c6">w</font></span>)</span> = 
<font color="Maroon" title="c4">z</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">v'</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c8">w'</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c7">v'</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c8">w'</font></span>)</span>
<b>by </b><i><a class="ref" href="clvect_1.html#D2" target="_self" title="CLVECT_1:def.2">Def2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c6">w</font></span>)</span>

;<br/>


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

<b>thus </b><a NAME="E2:125_1_1_1"/>
for <font color="Olive" title="b1">z1</font>, <font color="Olive" title="b2">z2</font> being  <a href="cfunct_1.html#NM1" title="CFUNCT_1:NM.1">Complex</a><br/> for <font color="Olive" title="b3">v</font> being  <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c3">X</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">z1</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <font color="Olive" title="b2">z2</font></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</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">z1</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Olive" title="b3">v</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">z2</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Olive" title="b3">v</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="125_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">z1</font>, <font color="Maroon" title="c5">z2</font> be   <a href="cfunct_1.html#NM1" title="CFUNCT_1:NM.1">Complex</a>;<br/><b>let </b><font color="Maroon" title="c6">v</font> be   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c3">X</font>;<br/>





<b>reconsider </b><font color="Maroon" title="c7">v'</font> = <font color="Maroon" title="c6">v</font> as   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="clvect_1.html#K2" title="CLVECT_1:func.2">(0).</a> <font color="Maroon" title="c1">V</font></span>)</span> ;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c4">z1</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <font color="Maroon" title="c5">z2</font></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c6">v</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">z1</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <font color="Maroon" title="c5">z2</font></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c7">v'</font>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">z1</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c7">v'</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">z2</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c7">v'</font></span>)</span>
<b>by </b><i><a class="ref" href="clvect_1.html#D2" target="_self" title="CLVECT_1:def.2">Def2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">z1</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c6">v</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">z2</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c6">v</font></span>)</span>

;<br/>


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

<b>thus </b><a NAME="E3:125_1_1_1"/>
for <font color="Olive" title="b1">z1</font>, <font color="Olive" title="b2">z2</font> being  <a href="cfunct_1.html#NM1" title="CFUNCT_1:NM.1">Complex</a><br/> for <font color="Olive" title="b3">v</font> being  <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c3">X</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">z1</font> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <font color="Olive" title="b2">z2</font></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Olive" title="b3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">z1</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">z2</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Olive" title="b3">v</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="125_1_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">z1</font>, <font color="Maroon" title="c5">z2</font> be   <a href="cfunct_1.html#NM1" title="CFUNCT_1:NM.1">Complex</a>;<br/><b>let </b><font color="Maroon" title="c6">v</font> be   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c3">X</font>;<br/>





<b>reconsider </b><font color="Maroon" title="c7">v'</font> = <font color="Maroon" title="c6">v</font> as   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="clvect_1.html#K2" title="CLVECT_1:func.2">(0).</a> <font color="Maroon" title="c1">V</font></span>)</span> ;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c4">z1</font> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <font color="Maroon" title="c5">z2</font></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c6">v</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">z1</font> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <font color="Maroon" title="c5">z2</font></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c7">v'</font>

<br/>.= 
<font color="Maroon" title="c4">z1</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">z2</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c7">v'</font></span>)</span>
<b>by </b><i><a class="ref" href="clvect_1.html#D2" target="_self" title="CLVECT_1:def.2">Def2</a></i>
<br/>.= 
<font color="Maroon" title="c4">z1</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">z2</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c6">v</font></span>)</span>

;<br/>


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

<b>let </b><font color="Maroon" title="c4">v</font> be   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c3">X</font>;<br/>

<b>reconsider </b><font color="Maroon" title="c5">v'</font> = <font color="Maroon" title="c4">v</font> as   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="clvect_1.html#K2" title="CLVECT_1:func.2">(0).</a> <font color="Maroon" title="c1">V</font></span>)</span> ;<br/>
<b>thus </b><a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a>  <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c4">v</font> = 
<a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a>  <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v'</font>

<br/>.= 
<font color="Maroon" title="c4">v</font>
<b>by </b><i><a class="ref" href="clvect_1.html#D2" target="_self" title="CLVECT_1:def.2">Def2</a></i>
;<br/>


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

<b>thus </b><a NAME="E4:125_1_1"/>
for <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> being  <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c3">X</font> holds  <font color="Olive" title="b1">v</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</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#K2" title="RLVECT_1:func.2">+</a> <font color="Olive" title="b1">v</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D5" title="RLVECT_1:def.5">RLVECT_1:def 5</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="125_1_1_2"><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="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c3">X</font>;<br/>

<b>reconsider </b><font color="Maroon" title="c6">v'</font> = <font color="Maroon" title="c4">v</font>, <font color="Maroon" title="c7">w'</font> = <font color="Maroon" title="c5">w</font> as   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="clvect_1.html#K2" title="CLVECT_1:func.2">(0).</a> <font color="Maroon" title="c1">V</font></span>)</span> ;<br/>
<b>thus </b><font color="Maroon" title="c4">v</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <font color="Maroon" title="c5">w</font> = 
<font color="Maroon" title="c7">w'</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c6">v'</font>
<b>by </b><i><a class="txt" href="clvect_1.html#E2:125_1_1"><i><font color="Green" title="E75">A1</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c5">w</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <font color="Maroon" title="c4">v</font>

;<br/>


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

<b>thus </b><a NAME="E5:125_1_1"/>
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="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c3">X</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">u</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#K2" title="RLVECT_1:func.2">+</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#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#K2" title="RLVECT_1:func.2">+</a> <font color="Olive" title="b3">w</font></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="125_1_1_3"><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="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c3">X</font>;<br/>

<b>reconsider </b><font color="Maroon" title="c7">u'</font> = <font color="Maroon" title="c4">u</font>, <font color="Maroon" title="c8">v'</font> = <font color="Maroon" title="c5">v</font>, <font color="Maroon" title="c9">w'</font> = <font color="Maroon" title="c6">w</font> as   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="clvect_1.html#K2" title="CLVECT_1:func.2">(0).</a> <font color="Maroon" title="c1">V</font></span>)</span> ;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c4">u</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#K2" title="RLVECT_1:func.2">+</a> <font color="Maroon" title="c6">w</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">u'</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c8">v'</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c9">w'</font>

<br/>.= 
<font color="Maroon" title="c7">u'</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">v'</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c9">w'</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#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#K2" title="RLVECT_1:func.2">+</a> <font color="Maroon" title="c6">w</font></span>)</span>

;<br/>


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

<b>thus </b><a NAME="E6:125_1_1"/>
for <font color="Olive" title="b1">v</font> being  <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c3">X</font> holds  <font color="Olive" title="b1">v</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D7" title="RLVECT_1:def.7">RLVECT_1:def 7</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="125_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">v</font> be   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c3">X</font>;<br/>

<b>reconsider </b><font color="Maroon" title="c5">v'</font> = <font color="Maroon" title="c4">v</font> as   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="clvect_1.html#K2" title="CLVECT_1:func.2">(0).</a> <font color="Maroon" title="c1">V</font></span>)</span> ;<br/>
<b>thus </b><font color="Maroon" title="c4">v</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c3">X</font></span>)</span> = 
<font color="Maroon" title="c5">v'</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <span class="p2">(<span class="default"><a href="clvect_1.html#K2" title="CLVECT_1:func.2">(0).</a> <font color="Maroon" title="c1">V</font></span>)</span></span>)</span>

<br/>.= 
<font color="Maroon" title="c4">v</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></i>
;<br/>


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

<b>thus </b><a NAME="E7:125_1_1"/>
for <font color="Olive" title="b1">v</font> being  <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c3">X</font>  ex <font color="Olive" title="b2">w</font> being  <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c3">X</font> st <font color="Olive" title="b1">v</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <font color="Olive" title="b2">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c3">X</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D8" title="RLVECT_1:def.8">RLVECT_1:def 8</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="125_1_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">v</font> be   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c3">X</font>;<br/>

<b>reconsider </b><font color="Maroon" title="c5">v'</font> = <font color="Maroon" title="c4">v</font> as   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="clvect_1.html#K2" title="CLVECT_1:func.2">(0).</a> <font color="Maroon" title="c1">V</font></span>)</span> ;<br/>
<b>consider </b><font color="Maroon" title="c6">w'</font> being   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="clvect_1.html#K2" title="CLVECT_1:func.2">(0).</a> <font color="Maroon" title="c1">V</font></span>)</span><b> such that </b><br/><a NAME="E3:125_1_1_5"/><i><font color="Green" title="E76">A2</font></i>: 
<font color="Maroon" title="c5">v'</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c6">w'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <span class="p1">(<span class="default"><a href="clvect_1.html#K2" title="CLVECT_1:func.2">(0).</a> <font color="Maroon" title="c1">V</font></span>)</span>
 <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">w'</font> as   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c3">X</font> ;<br/>
<b>take </b>
<font color="Maroon" title="c7">w</font>
;<br/>

<b>thus </b><a NAME="E5:125_1_1_5"/>
<font color="Maroon" title="c4">v</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <font color="Maroon" title="c7">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c3">X</font>
 <b>by </b><i><a class="txt" href="clvect_1.html#E3:125_1_1_5"><i><font color="Green" title="E76">A2</font></i></a></i>;<br/>


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

<b>thus </b><a NAME="E8:125_1_1"/>
<font color="Maroon" title="c3">X</font> is <a href="clvect_1.html#V4" title="CLVECT_1:attr.4">strict</a>
 ;<br/>


</div>
