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

<b>set </b><font color="Maroon">W</font> =  <a href="clvect_1.html#G1" target="_self">CLSStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,the <a href="struct_0.html#U2">Zero</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="clvect_1.html#U1" target="_self">Mult</a> of <font color="Maroon">V</font> #);<br/>
<a NAME="E1:70_1_1"/>
(  <a href="clvect_1.html#G1" target="_self">CLSStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,the <a href="struct_0.html#U2">Zero</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="clvect_1.html#U1" target="_self">Mult</a> of <font color="Maroon">V</font> #) is <a href="rlvect_1.html#V3">Abelian</a> &amp;  <a href="clvect_1.html#G1" target="_self">CLSStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,the <a href="struct_0.html#U2">Zero</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="clvect_1.html#U1" target="_self">Mult</a> of <font color="Maroon">V</font> #) is <a href="rlvect_1.html#V4">add-associative</a> &amp;  <a href="clvect_1.html#G1" target="_self">CLSStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,the <a href="struct_0.html#U2">Zero</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="clvect_1.html#U1" target="_self">Mult</a> of <font color="Maroon">V</font> #) is <a href="rlvect_1.html#V5">right_zeroed</a> &amp;  <a href="clvect_1.html#G1" target="_self">CLSStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,the <a href="struct_0.html#U2">Zero</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="clvect_1.html#U1" target="_self">Mult</a> of <font color="Maroon">V</font> #) is <a href="rlvect_1.html#V6">right_complementable</a> &amp;  <a href="clvect_1.html#G1" target="_self">CLSStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,the <a href="struct_0.html#U2">Zero</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="clvect_1.html#U1" target="_self">Mult</a> of <font color="Maroon">V</font> #) is <a href="clvect_1.html#V2" target="_self">ComplexLinearSpace-like</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:70_1_1_1"/><i><font color="Green">E28</font></i>: 
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 <a href="clvect_1.html#G1" target="_self">CLSStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,the <a href="struct_0.html#U2">Zero</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="clvect_1.html#U1" target="_self">Mult</a> of <font color="Maroon">V</font> #)<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">V</font>  st <font color="Olive">v</font> <a href="hidden.html#R1">=</a> <font color="Olive">v'</font> &amp; <font color="Olive">w</font> <a href="hidden.html#R1">=</a> <font color="Olive">w'</font> holds <br/>( <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">v'</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">w'</font> &amp; <font color="Olive">z</font> <a href="clvect_1.html#K1" target="_self">*</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a> <font color="Olive">z</font> <a href="clvect_1.html#K1" target="_self">*</a> <font color="Olive">v'</font> )
 
;<br/>
<b>thus </b><a NAME="E2:70_1_1_1"/>
for <font color="Olive">v</font>, <font color="Olive">w</font> being  <a href="clvect_1.html#NM1">VECTOR</a> of <a href="clvect_1.html#G1" target="_self">CLSStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,the <a href="struct_0.html#U2">Zero</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="clvect_1.html#U1" target="_self">Mult</a> of <font color="Maroon">V</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="70_1_1_1_1"><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 <a href="clvect_1.html#G1" target="_self">CLSStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,the <a href="struct_0.html#U2">Zero</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="clvect_1.html#U1" target="_self">Mult</a> of <font color="Maroon">V</font> #), <font color="Maroon">w</font> be   <a href="clvect_1.html#NM1">VECTOR</a> of <a href="clvect_1.html#G1" target="_self">CLSStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,the <a href="struct_0.html#U2">Zero</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="clvect_1.html#U1" target="_self">Mult</a> of <font color="Maroon">V</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 <font color="Maroon">V</font> ;<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="E3:70_1_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 <a href="clvect_1.html#G1" target="_self">CLSStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,the <a href="struct_0.html#U2">Zero</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="clvect_1.html#U1" target="_self">Mult</a> of <font color="Maroon">V</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="70_1_1_1_2"><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 <a href="clvect_1.html#G1" target="_self">CLSStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,the <a href="struct_0.html#U2">Zero</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="clvect_1.html#U1" target="_self">Mult</a> of <font color="Maroon">V</font> #), <font color="Maroon">v</font> be   <a href="clvect_1.html#NM1">VECTOR</a> of <a href="clvect_1.html#G1" target="_self">CLSStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,the <a href="struct_0.html#U2">Zero</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="clvect_1.html#U1" target="_self">Mult</a> of <font color="Maroon">V</font> #), <font color="Maroon">w</font> be   <a href="clvect_1.html#NM1">VECTOR</a> of <a href="clvect_1.html#G1" target="_self">CLSStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,the <a href="struct_0.html#U2">Zero</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="clvect_1.html#U1" target="_self">Mult</a> of <font color="Maroon">V</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 <font color="Maroon">V</font> ;<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="E4:70_1_1_1"/>
for <font color="Olive">v</font> being  <a href="clvect_1.html#NM1">VECTOR</a> of <a href="clvect_1.html#G1" target="_self">CLSStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,the <a href="struct_0.html#U2">Zero</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="clvect_1.html#U1" target="_self">Mult</a> of <font color="Maroon">V</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> <a href="clvect_1.html#G1" target="_self">CLSStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,the <a href="struct_0.html#U2">Zero</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="clvect_1.html#U1" target="_self">Mult</a> of <font color="Maroon">V</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="70_1_1_1_3"><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 <a href="clvect_1.html#G1" target="_self">CLSStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,the <a href="struct_0.html#U2">Zero</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="clvect_1.html#U1" target="_self">Mult</a> of <font color="Maroon">V</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 <font color="Maroon">V</font> ;<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> <a href="clvect_1.html#G1" target="_self">CLSStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,the <a href="struct_0.html#U2">Zero</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="clvect_1.html#U1" target="_self">Mult</a> of <font color="Maroon">V</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> <font color="Maroon">V</font></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="E5:70_1_1_1"/>
for <font color="Olive">v</font> being  <a href="clvect_1.html#NM1">VECTOR</a> of <a href="clvect_1.html#G1" target="_self">CLSStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,the <a href="struct_0.html#U2">Zero</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="clvect_1.html#U1" target="_self">Mult</a> of <font color="Maroon">V</font> #)  ex <font color="Olive">w</font> being  <a href="clvect_1.html#NM1">VECTOR</a> of <a href="clvect_1.html#G1" target="_self">CLSStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,the <a href="struct_0.html#U2">Zero</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="clvect_1.html#U1" target="_self">Mult</a> of <font color="Maroon">V</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> <a href="clvect_1.html#G1" target="_self">CLSStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,the <a href="struct_0.html#U2">Zero</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="clvect_1.html#U1" target="_self">Mult</a> of <font color="Maroon">V</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="70_1_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 <a href="clvect_1.html#G1" target="_self">CLSStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,the <a href="struct_0.html#U2">Zero</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="clvect_1.html#U1" target="_self">Mult</a> of <font color="Maroon">V</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 <font color="Maroon">V</font> ;<br/>
<b>consider </b><font color="Maroon">w'</font> being   <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E3:70_1_1_1_4"/><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> <font color="Maroon">V</font>
 <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 <a href="clvect_1.html#G1" target="_self">CLSStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,the <a href="struct_0.html#U2">Zero</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="clvect_1.html#U1" target="_self">Mult</a> of <font color="Maroon">V</font> #) ;<br/>
<b>take </b>
<font color="Maroon">w</font>
;<br/>

<b>thus </b><a NAME="E5:70_1_1_1_4"/>
<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> <a href="clvect_1.html#G1" target="_self">CLSStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,the <a href="struct_0.html#U2">Zero</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="clvect_1.html#U1" target="_self">Mult</a> of <font color="Maroon">V</font> #)
 <b>by </b><i/>;<br/>


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

<b>thus </b><a NAME="E6:70_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 <a href="clvect_1.html#G1" target="_self">CLSStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,the <a href="struct_0.html#U2">Zero</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="clvect_1.html#U1" target="_self">Mult</a> of <font color="Maroon">V</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="70_1_1_1_5"><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 <a href="clvect_1.html#G1" target="_self">CLSStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,the <a href="struct_0.html#U2">Zero</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="clvect_1.html#U1" target="_self">Mult</a> of <font color="Maroon">V</font> #), <font color="Maroon">w</font> be   <a href="clvect_1.html#NM1">VECTOR</a> of <a href="clvect_1.html#G1" target="_self">CLSStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,the <a href="struct_0.html#U2">Zero</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="clvect_1.html#U1" target="_self">Mult</a> of <font color="Maroon">V</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 <font color="Maroon">V</font> ;<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="E7:70_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 <a href="clvect_1.html#G1" target="_self">CLSStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,the <a href="struct_0.html#U2">Zero</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="clvect_1.html#U1" target="_self">Mult</a> of <font color="Maroon">V</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="70_1_1_1_6"><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 <a href="clvect_1.html#G1" target="_self">CLSStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,the <a href="struct_0.html#U2">Zero</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="clvect_1.html#U1" target="_self">Mult</a> of <font color="Maroon">V</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 <font color="Maroon">V</font> ;<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="E8:70_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 <a href="clvect_1.html#G1" target="_self">CLSStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,the <a href="struct_0.html#U2">Zero</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="clvect_1.html#U1" target="_self">Mult</a> of <font color="Maroon">V</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="70_1_1_1_7"><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 <a href="clvect_1.html#G1" target="_self">CLSStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,the <a href="struct_0.html#U2">Zero</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="clvect_1.html#U1" target="_self">Mult</a> of <font color="Maroon">V</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 <font color="Maroon">V</font> ;<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>thus </b><a NAME="E9:70_1_1_1"/>
for <font color="Olive">v</font> being  <a href="clvect_1.html#NM1">VECTOR</a> of <a href="clvect_1.html#G1" target="_self">CLSStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,the <a href="struct_0.html#U2">Zero</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="clvect_1.html#U1" target="_self">Mult</a> of <font color="Maroon">V</font> #) holds  <a href="complex1.html#K6">1r</a>  <a href="clvect_1.html#K1" target="_self">*</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a> <font color="Olive">v</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_1_1_8"><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 <a href="clvect_1.html#G1" target="_self">CLSStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,the <a href="struct_0.html#U2">Zero</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="clvect_1.html#U1" target="_self">Mult</a> of <font color="Maroon">V</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 <font color="Maroon">V</font> ;<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>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">W</font> =  <a href="clvect_1.html#G1" target="_self">CLSStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,the <a href="struct_0.html#U2">Zero</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="clvect_1.html#U1" target="_self">Mult</a> of <font color="Maroon">V</font> #) as   <a href="clvect_1.html#NM2">ComplexLinearSpace</a> ;<br/>
<a NAME="E3:70_1_1"/>
<font color="Maroon">W</font> is    <a href="clvect_1.html#M1" target="_self">Subspace</a> of <font color="Maroon">V</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_1_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:70_1_1_2"/>
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> &amp; the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">W</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">V</font> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="clvect_1.html#D4" target="_self">CLVECT_1:def 4</a><br/>

<b>thus </b><a NAME="E2:70_1_1_2"/>
( the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">W</font> <a href="hidden.html#R1">=</a> the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">V</font> <a href="realset1.html#K1">||</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W</font> &amp; the <a href="clvect_1.html#U1" target="_self">Mult</a> of <font color="Maroon">W</font> <a href="hidden.html#R1">=</a> the <a href="clvect_1.html#U1" target="_self">Mult</a> of <font color="Maroon">V</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K2">COMPLEX</a> ,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W</font></span><a href="zfmisc_1.html#K2">:]</a></span> )
 <b>by </b><i><a class="ref" href="funct_2.html#T40">FUNCT_2:40</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:70_1_1"/>
 <a href="clvect_1.html#G1" target="_self">CLSStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,the <a href="struct_0.html#U2">Zero</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="clvect_1.html#U1" target="_self">Mult</a> of <font color="Maroon">V</font> #) is  <a href="clvect_1.html#V1" target="_self">strict</a>  <a href="clvect_1.html#M1" target="_self">Subspace</a> of <font color="Maroon">V</font>
 ;<br/>


</div>
