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

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

<b>thus </b><a NAME="E1:125_1_1"/>
<font color="Maroon">X</font> is <a href="clvect_1.html#V5" target="_self">ComplexNormSpace-like</a>
 <b>by </b><i>, </i>;<br/>

<a NAME="E2:125_1_1"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font><br/> for <font color="Olive">x'</font>, <font color="Olive">y'</font> being  <a href="clvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="clvect_1.html#K2" target="_self">(0).</a> <font color="Maroon">V</font></span>)</span>  st <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">x'</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">y'</font> holds <br/>( <font color="Olive">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">x'</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">y'</font> &amp; ( for <font color="Olive">z</font> being  <a href="cfunct_1.html#NM1">Complex</a> holds  <font color="Olive">z</font> <a href="clvect_1.html#K1" target="_self">*</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">z</font> <a href="clvect_1.html#K1" target="_self">*</a> <font color="Olive">x'</font> ) )
 
;<br/>
<b>thus </b><a NAME="E3:125_1_1"/>
<font color="Maroon">X</font> is <a href="clvect_1.html#V2" target="_self">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">z</font> being  <a href="cfunct_1.html#NM1">Complex</a><br/> for <font color="Olive">v</font>, <font color="Olive">w</font> being  <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font> holds  <font color="Olive">z</font> <a href="clvect_1.html#K1" target="_self">*</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">z</font> <a href="clvect_1.html#K1" target="_self">*</a> <font color="Olive">v</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Olive">z</font> <a href="clvect_1.html#K1" target="_self">*</a> <font color="Olive">w</font></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="clvect_1.html#D2" target="_self">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">z</font> be   <a href="cfunct_1.html#NM1">Complex</a>;<br/><b>let </b><font color="Maroon">v</font> be   <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font>, <font color="Maroon">w</font> be   <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</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="clvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="clvect_1.html#K2" target="_self">(0).</a> <font color="Maroon">V</font></span>)</span> ;<br/>
<b>thus </b><font color="Maroon">z</font> <a href="clvect_1.html#K1" target="_self">*</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> = 
<font color="Maroon">z</font> <a href="clvect_1.html#K1" target="_self">*</a> <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>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="clvect_1.html#K1" target="_self">*</a> <font color="Maroon">v'</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="clvect_1.html#K1" target="_self">*</a> <font color="Maroon">w'</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="clvect_1.html#K1" target="_self">*</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="clvect_1.html#K1" target="_self">*</a> <font color="Maroon">w</font></span>)</span>

;<br/>


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

<b>thus </b><a NAME="E2:125_1_1_1"/>
for <font color="Olive">z1</font>, <font color="Olive">z2</font> being  <a href="cfunct_1.html#NM1">Complex</a><br/> for <font color="Olive">v</font> being  <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font> holds  <span class="p1">(<span class="default"><font color="Olive">z1</font> <a href="complex1.html#K8">+</a> <font color="Olive">z2</font></span>)</span> <a href="clvect_1.html#K1" target="_self">*</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">z1</font> <a href="clvect_1.html#K1" target="_self">*</a> <font color="Olive">v</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Olive">z2</font> <a href="clvect_1.html#K1" target="_self">*</a> <font color="Olive">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">z1</font> be   <a href="cfunct_1.html#NM1">Complex</a>;<br/><b>let </b><font color="Maroon">z2</font> be   <a href="cfunct_1.html#NM1">Complex</a>;<br/><b>let </b><font color="Maroon">v</font> be   <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font>;<br/>





<b>reconsider </b><font color="Maroon">v'</font> = <font color="Maroon">v</font> as   <a href="clvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="clvect_1.html#K2" target="_self">(0).</a> <font color="Maroon">V</font></span>)</span> ;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">z1</font> <a href="complex1.html#K8">+</a> <font color="Maroon">z2</font></span>)</span> <a href="clvect_1.html#K1" target="_self">*</a> <font color="Maroon">v</font> = 
<span class="p1">(<span class="default"><font color="Maroon">z1</font> <a href="complex1.html#K8">+</a> <font color="Maroon">z2</font></span>)</span> <a href="clvect_1.html#K1" target="_self">*</a> <font color="Maroon">v'</font>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">z1</font> <a href="clvect_1.html#K1" target="_self">*</a> <font color="Maroon">v'</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">z2</font> <a href="clvect_1.html#K1" target="_self">*</a> <font color="Maroon">v'</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">z1</font> <a href="clvect_1.html#K1" target="_self">*</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">z2</font> <a href="clvect_1.html#K1" target="_self">*</a> <font color="Maroon">v</font></span>)</span>

;<br/>


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

<b>thus </b><a NAME="E3:125_1_1_1"/>
for <font color="Olive">z1</font>, <font color="Olive">z2</font> being  <a href="cfunct_1.html#NM1">Complex</a><br/> for <font color="Olive">v</font> being  <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font> holds  <span class="p1">(<span class="default"><font color="Olive">z1</font> <a href="complex1.html#K9">*</a> <font color="Olive">z2</font></span>)</span> <a href="clvect_1.html#K1" target="_self">*</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a> <font color="Olive">z1</font> <a href="clvect_1.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><font color="Olive">z2</font> <a href="clvect_1.html#K1" target="_self">*</a> <font color="Olive">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">z1</font> be   <a href="cfunct_1.html#NM1">Complex</a>;<br/><b>let </b><font color="Maroon">z2</font> be   <a href="cfunct_1.html#NM1">Complex</a>;<br/><b>let </b><font color="Maroon">v</font> be   <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font>;<br/>





<b>reconsider </b><font color="Maroon">v'</font> = <font color="Maroon">v</font> as   <a href="clvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="clvect_1.html#K2" target="_self">(0).</a> <font color="Maroon">V</font></span>)</span> ;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">z1</font> <a href="complex1.html#K9">*</a> <font color="Maroon">z2</font></span>)</span> <a href="clvect_1.html#K1" target="_self">*</a> <font color="Maroon">v</font> = 
<span class="p1">(<span class="default"><font color="Maroon">z1</font> <a href="complex1.html#K9">*</a> <font color="Maroon">z2</font></span>)</span> <a href="clvect_1.html#K1" target="_self">*</a> <font color="Maroon">v'</font>

<br/>.= 
<font color="Maroon">z1</font> <a href="clvect_1.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">z2</font> <a href="clvect_1.html#K1" target="_self">*</a> <font color="Maroon">v'</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">z1</font> <a href="clvect_1.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">z2</font> <a href="clvect_1.html#K1" target="_self">*</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="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font>;<br/>

<b>reconsider </b><font color="Maroon">v'</font> = <font color="Maroon">v</font> as   <a href="clvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="clvect_1.html#K2" target="_self">(0).</a> <font color="Maroon">V</font></span>)</span> ;<br/>
<b>thus </b><a href="complex1.html#K6">1r</a>  <a href="clvect_1.html#K1" target="_self">*</a> <font color="Maroon">v</font> = 
<a href="complex1.html#K6">1r</a>  <a href="clvect_1.html#K1" target="_self">*</a> <font color="Maroon">v'</font>

<br/>.= 
<font color="Maroon">v</font>
<b>by </b><i/>
;<br/>


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

<b>thus </b><a NAME="E4:125_1_1"/>
for <font color="Olive">v</font>, <font color="Olive">w</font> being  <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font> holds  <font color="Olive">v</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">w</font> <a href="hidden.html#R1">=</a> <font color="Olive">w</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">v</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D5">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">v</font> be   <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font>, <font color="Maroon">w</font> be   <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</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="clvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="clvect_1.html#K2" target="_self">(0).</a> <font color="Maroon">V</font></span>)</span> ;<br/>
<b>thus </b><font color="Maroon">v</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">w</font> = 
<font color="Maroon">w'</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v'</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">w</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">v</font>

;<br/>


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

<b>thus </b><a NAME="E5:125_1_1"/>
for <font color="Olive">u</font>, <font color="Olive">v</font>, <font color="Olive">w</font> being  <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font> holds  <span class="p1">(<span class="default"><font color="Olive">u</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">v</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Olive">w</font> <a href="hidden.html#R1">=</a> <font color="Olive">u</font> <a href="rlvect_1.html#K2">+</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>
  <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D6">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">u</font> be   <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font>, <font color="Maroon">v</font> be   <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font>, <font color="Maroon">w</font> be   <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font>;<br/>

<b>reconsider </b><font color="Maroon">u'</font> = <font color="Maroon">u</font>, <font color="Maroon">v'</font> = <font color="Maroon">v</font>, <font color="Maroon">w'</font> = <font color="Maroon">w</font> as   <a href="clvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="clvect_1.html#K2" target="_self">(0).</a> <font color="Maroon">V</font></span>)</span> ;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">w</font> = 
<span class="p1">(<span class="default"><font color="Maroon">u'</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v'</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">w'</font>

<br/>.= 
<font color="Maroon">u'</font> <a href="rlvect_1.html#K4">+</a> <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>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
<font color="Maroon">u</font> <a href="rlvect_1.html#K2">+</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>

;<br/>


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

<b>thus </b><a NAME="E6:125_1_1"/>
for <font color="Olive">v</font> being  <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font> holds  <font color="Olive">v</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">v</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D7">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">v</font> be   <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font>;<br/>

<b>reconsider </b><font color="Maroon">v'</font> = <font color="Maroon">v</font> as   <a href="clvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="clvect_1.html#K2" target="_self">(0).</a> <font color="Maroon">V</font></span>)</span> ;<br/>
<b>thus </b><font color="Maroon">v</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font></span>)</span> = 
<font color="Maroon">v'</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <span class="p2">(<span class="default"><a href="clvect_1.html#K2" target="_self">(0).</a> <font color="Maroon">V</font></span>)</span></span>)</span>

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


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

<b>thus </b><a NAME="E7:125_1_1"/>
for <font color="Olive">v</font> being  <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font>  ex <font color="Olive">w</font> being  <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font> st <font color="Olive">v</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">w</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D8">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">v</font> be   <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font>;<br/>

<b>reconsider </b><font color="Maroon">v'</font> = <font color="Maroon">v</font> as   <a href="clvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="clvect_1.html#K2" target="_self">(0).</a> <font color="Maroon">V</font></span>)</span> ;<br/>
<b>consider </b><font color="Maroon">w'</font> being   <a href="clvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="clvect_1.html#K2" target="_self">(0).</a> <font color="Maroon">V</font></span>)</span><b> such that </b><br/><a NAME="E3:125_1_1_5"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">v'</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">w'</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="clvect_1.html#K2" target="_self">(0).</a> <font color="Maroon">V</font></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D8">RLVECT_1:def 8</a></i>;<br/>
<b>reconsider </b><font color="Maroon">w</font> = <font color="Maroon">w'</font> as   <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font> ;<br/>
<b>take </b>
<font color="Maroon">w</font>
;<br/>

<b>thus </b><a NAME="E5:125_1_1_5"/>
<font color="Maroon">v</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">w</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font>
 <b>by </b><i/>;<br/>


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

<b>thus </b><a NAME="E8:125_1_1"/>
<font color="Maroon">X</font> is <a href="clvect_1.html#V4" target="_self">strict</a>
 ;<br/>


</div>
