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

<b>let </b><font color="Maroon" title="c1">V</font> be   <a href="clvect_1.html#NM2" title="CLVECT_1:NM.2">ComplexLinearSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">V1</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font>;<br/><b>let </b><font color="Maroon" title="c3">D</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c4">d1</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">D</font>;<br/><b>let </b><font color="Maroon" title="c5">A</font> be   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c3">D</font>;<br/><b>let </b><font color="Maroon" title="c6">M</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> ,<font color="Maroon" title="c3">D</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c3">D</font>;<br/>











<b>assume </b><b>that </b><br/><a NAME="E1:57"/><i><font color="Green" title="E32">A1</font></i>: 
<font color="Maroon" title="c2">V1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">D</font>
 <b>and </b><br/><a NAME="E2:57"/><i><font color="Green" title="E33">A2</font></i>: 
<font color="Maroon" title="c4">d1</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="c1">V</font>
 <b>and </b><br/><a NAME="E3:57"/><i><font color="Green" title="E34">A3</font></i>: 
<font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">V</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c2">V1</font>
 <b>and </b><br/><a NAME="E4:57"/><i><font color="Green" title="E35">A4</font></i>: 
<font color="Maroon" title="c6">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="clvect_1.html#U1" title="CLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c1">V</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> ,<font color="Maroon" title="c2">V1</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ;<br/>

<b>set </b><font color="Maroon" title="c7">W</font> =  <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">d1</font>,<font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">M</font> #);<br/>
<a NAME="E5:57"/><i><font color="Green" title="E36">A5</font></i>: 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">d1</font>,<font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">M</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="c1">V</font>
 
<b>by </b><i><a class="txt" href="clvect_1.html#E2:57"><i><font color="Green" title="E33">A2</font></i></a></i>;<br/>
<a NAME="E6:57"/><i><font color="Green" title="E37">A6</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 <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">d1</font>,<font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">M</font> #) holds  <font color="Olive" title="b1">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 
<b>by </b><i><a class="txt" href="clvect_1.html#E1:57"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="clvect_1.html#E3:57"><i><font color="Green" title="E34">A3</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/>
<a NAME="E7:57"/>
(  <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">d1</font>,<font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">M</font> #) is <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a> &amp;  <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">d1</font>,<font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">M</font> #) is <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a> &amp;  <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">d1</font>,<font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">M</font> #) is <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a> &amp;  <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">d1</font>,<font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">M</font> #) is <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a> &amp;  <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">d1</font>,<font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">M</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="57_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon" title="c8">AV</font> = the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">V</font>;<br/>
<b>set </b><font color="Maroon" title="c9">MV</font> = the <a href="clvect_1.html#U1" title="CLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c1">V</font>;<br/>
<b>thus </b><a NAME="E1:57_1"/>
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 <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">d1</font>,<font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">M</font> #) holds  <font color="Olive" title="b1">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b1">x</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="57_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">x</font>, <font color="Maroon" title="c11">y</font> be   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">d1</font>,<font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">M</font> #);<br/>

<b>reconsider </b><font color="Maroon" title="c12">x1</font> = <font color="Maroon" title="c10">x</font>, <font color="Maroon" title="c13">y1</font> = <font color="Maroon" title="c11">y</font> as   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="txt" href="clvect_1.html#E1:57"><i><font color="Green" title="E32">A1</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c10">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c11">y</font> = 
<font color="Maroon" title="c12">x1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c13">y1</font>
<b>by </b><i><a class="txt" href="clvect_1.html#E6:57"><i><font color="Green" title="E37">A6</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c13">y1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c12">x1</font>

<br/>.= 
<font color="Maroon" title="c11">y</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c10">x</font>
<b>by </b><i><a class="txt" href="clvect_1.html#E6:57"><i><font color="Green" title="E37">A6</font></i></a></i>
;<br/>


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

<b>thus </b><a NAME="E2:57_1"/>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</font> being  <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">d1</font>,<font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">M</font> #) holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">y</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b3">z</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="57_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">x</font>, <font color="Maroon" title="c11">y</font>, <font color="Maroon" title="c12">z</font> be   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">d1</font>,<font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">M</font> #);<br/>

<b>reconsider </b><font color="Maroon" title="c13">x1</font> = <font color="Maroon" title="c10">x</font>, <font color="Maroon" title="c14">y1</font> = <font color="Maroon" title="c11">y</font>, <font color="Maroon" title="c15">z1</font> = <font color="Maroon" title="c12">z</font> as   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="txt" href="clvect_1.html#E1:57"><i><font color="Green" title="E32">A1</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c10">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c11">y</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c12">z</font> = 
the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c11">y</font></span>)</span>,<font color="Maroon" title="c15">z1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<b>by </b><i><a class="txt" href="clvect_1.html#E6:57"><i><font color="Green" title="E37">A6</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">x1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c14">y1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c15">z1</font>
<b>by </b><i><a class="txt" href="clvect_1.html#E6:57"><i><font color="Green" title="E37">A6</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c13">x1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">y1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c15">z1</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/>.= 
the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c13">x1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">y</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c12">z</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<b>by </b><i><a class="txt" href="clvect_1.html#E6:57"><i><font color="Green" title="E37">A6</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c10">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">y</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c12">z</font></span>)</span>
<b>by </b><i><a class="txt" href="clvect_1.html#E6:57"><i><font color="Green" title="E37">A6</font></i></a></i>
;<br/>


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

<b>thus </b><a NAME="E3:57_1"/>
for <font color="Olive" title="b1">x</font> being  <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">d1</font>,<font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">M</font> #) holds  <font color="Olive" title="b1">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">d1</font>,<font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">M</font> #)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</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="57_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">x</font> be   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">d1</font>,<font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">M</font> #);<br/>

<b>reconsider </b><font color="Maroon" title="c11">y</font> = <font color="Maroon" title="c10">x</font> as   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="txt" href="clvect_1.html#E1:57"><i><font color="Green" title="E32">A1</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c10">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">d1</font>,<font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">M</font> #)</span>)</span> = 
<font color="Maroon" title="c11">y</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="c1">V</font></span>)</span>
<b>by </b><i><a class="txt" href="clvect_1.html#E2:57"><i><font color="Green" title="E33">A2</font></i></a>, <a class="txt" href="clvect_1.html#E6:57"><i><font color="Green" title="E37">A6</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c10">x</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="E4:57_1"/>
 <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">d1</font>,<font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">M</font> #) is <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">x</font> be   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">d1</font>,<font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">M</font> #); <i><font color="Red">:: according to </font></i><a class="ref" href="algstr_0.html#D16" title="ALGSTR_0:def.16">ALGSTR_0:def 16</a><br/>

<b>reconsider </b><font color="Maroon" title="c11">x1</font> = <font color="Maroon" title="c10">x</font> as   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="txt" href="clvect_1.html#E1:57"><i><font color="Green" title="E32">A1</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">v</font> being   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E3:57_1_4"/><i><font color="Green" title="E38">A7</font></i>: 
<font color="Maroon" title="c11">x1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c12">v</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="c1">V</font>
 <b>by </b><i><a class="ref" href="algstr_0.html#D11" title="ALGSTR_0:def.11">ALGSTR_0:def 11</a></i>;<br/>
<font color="Maroon" title="c12">v</font> = 
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c11">x1</font>
<b>by </b><i><a class="txt" href="clvect_1.html#E3:57_1_4"><i><font color="Green" title="E38">A7</font></i></a>, <a class="ref" href="rlvect_1.html#D11" title="RLVECT_1:def.11">RLVECT_1:def 11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a> </span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c11">x1</font>
<b>by </b><i><a class="ref" href="clvect_1.html#T4" target="_self" title="CLVECT_1:th.4">Th4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a> </span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c10">x</font>
<b>by </b><i><a class="txt" href="clvect_1.html#E1:57"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="clvect_1.html#E4:57"><i><font color="Green" title="E35">A4</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
;<br/>
<b>then reconsider </b><font color="Maroon" title="c13">y</font> = <font color="Maroon" title="c12">v</font> as   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">d1</font>,<font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">M</font> #) ;<br/>
<b>take </b>
<font color="Maroon" title="c13">y</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="algstr_0.html#D11" title="ALGSTR_0:def.11">ALGSTR_0:def 11</a><br/>

<b>thus </b><a NAME="E6:57_1_4"/>
<font color="Maroon" title="c10">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c13">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">d1</font>,<font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">M</font> #)
 <b>by </b><i><a class="txt" href="clvect_1.html#E2:57"><i><font color="Green" title="E33">A2</font></i></a>, <a class="txt" href="clvect_1.html#E6:57"><i><font color="Green" title="E37">A6</font></i></a>, <a class="txt" href="clvect_1.html#E3:57_1_4"><i><font color="Green" title="E38">A7</font></i></a></i>;<br/>


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

<b>thus </b><a NAME="E5:57_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">x</font>, <font color="Olive" title="b3">y</font> being  <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">d1</font>,<font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">M</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">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b3">y</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">x</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.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="b3">y</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="57_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">z</font> be   <a href="cfunct_1.html#NM1" title="CFUNCT_1:NM.1">Complex</a>;<br/><b>let </b><font color="Maroon" title="c11">x</font>, <font color="Maroon" title="c12">y</font> be   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">d1</font>,<font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">M</font> #);<br/>



<b>reconsider </b><font color="Maroon" title="c13">x1</font> = <font color="Maroon" title="c11">x</font>, <font color="Maroon" title="c14">y1</font> = <font color="Maroon" title="c12">y</font> as   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="txt" href="clvect_1.html#E1:57"><i><font color="Green" title="E32">A1</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c10">z</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c12">y</font></span>)</span> = 
the <a href="clvect_1.html#U1" title="CLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c1">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c10">z</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c12">y</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<b>by </b><i><a class="txt" href="clvect_1.html#E1:57"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="clvect_1.html#E4:57"><i><font color="Green" title="E35">A4</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon" title="c10">z</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">x1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c14">y1</font></span>)</span>
<b>by </b><i><a class="txt" href="clvect_1.html#E6:57"><i><font color="Green" title="E37">A6</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c13">x1</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="c10">z</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c14">y1</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/>.= 
the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default">the <a href="clvect_1.html#U1" title="CLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c1">V</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c10">z</font>,<font color="Maroon" title="c13">x1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c12">y</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<b>by </b><i><a class="txt" href="clvect_1.html#E1:57"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="clvect_1.html#E4:57"><i><font color="Green" title="E35">A4</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c11">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c12">y</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<b>by </b><i><a class="txt" href="clvect_1.html#E1:57"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="clvect_1.html#E4:57"><i><font color="Green" title="E35">A4</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c12">y</font></span>)</span>
<b>by </b><i><a class="txt" href="clvect_1.html#E1:57"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="clvect_1.html#E3:57"><i><font color="Green" title="E34">A3</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
;<br/>


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

<b>thus </b><a NAME="E6:57_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">x</font> being  <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">d1</font>,<font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">M</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">x</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">x</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0: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">x</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">z1</font>, <font color="Maroon" title="c11">z2</font> be   <a href="cfunct_1.html#NM1" title="CFUNCT_1:NM.1">Complex</a>;<br/><b>let </b><font color="Maroon" title="c12">x</font> be   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">d1</font>,<font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">M</font> #);<br/>





<b>reconsider </b><font color="Maroon" title="c13">y</font> = <font color="Maroon" title="c12">x</font> as   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="txt" href="clvect_1.html#E1:57"><i><font color="Green" title="E32">A1</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c10">z1</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <font color="Maroon" title="c11">z2</font></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c12">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">z1</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <font color="Maroon" title="c11">z2</font></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c13">y</font>
<b>by </b><i><a class="txt" href="clvect_1.html#E1:57"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="clvect_1.html#E4:57"><i><font color="Green" title="E35">A4</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">z1</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c13">y</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="c11">z2</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c13">y</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/>.= 
the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default">the <a href="clvect_1.html#U1" title="CLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c1">V</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c10">z1</font>,<font color="Maroon" title="c13">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">z2</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c12">x</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<b>by </b><i><a class="txt" href="clvect_1.html#E1:57"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="clvect_1.html#E4:57"><i><font color="Green" title="E35">A4</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">z1</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c12">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">z2</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c12">x</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<b>by </b><i><a class="txt" href="clvect_1.html#E1:57"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="clvect_1.html#E4:57"><i><font color="Green" title="E35">A4</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">z1</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c12">x</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">z2</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c12">x</font></span>)</span>
<b>by </b><i><a class="txt" href="clvect_1.html#E1:57"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="clvect_1.html#E3:57"><i><font color="Green" title="E34">A3</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
;<br/>


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

<b>thus </b><a NAME="E7:57_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">x</font> being  <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">d1</font>,<font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">M</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">x</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">x</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">z1</font>, <font color="Maroon" title="c11">z2</font> be   <a href="cfunct_1.html#NM1" title="CFUNCT_1:NM.1">Complex</a>;<br/><b>let </b><font color="Maroon" title="c12">x</font> be   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">d1</font>,<font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">M</font> #);<br/>





<b>reconsider </b><font color="Maroon" title="c13">y</font> = <font color="Maroon" title="c12">x</font> as   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="txt" href="clvect_1.html#E1:57"><i><font color="Green" title="E32">A1</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c10">z1</font> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <font color="Maroon" title="c11">z2</font></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c12">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">z1</font> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <font color="Maroon" title="c11">z2</font></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c13">y</font>
<b>by </b><i><a class="txt" href="clvect_1.html#E1:57"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="clvect_1.html#E4:57"><i><font color="Green" title="E35">A4</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon" title="c10">z1</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">z2</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c13">y</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/>.= 
the <a href="clvect_1.html#U1" title="CLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c1">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c10">z1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">z2</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c12">x</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<b>by </b><i><a class="txt" href="clvect_1.html#E1:57"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="clvect_1.html#E4:57"><i><font color="Green" title="E35">A4</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon" title="c10">z1</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">z2</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c12">x</font></span>)</span>
<b>by </b><i><a class="txt" href="clvect_1.html#E1:57"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="clvect_1.html#E4:57"><i><font color="Green" title="E35">A4</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
;<br/>


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

<b>let </b><font color="Maroon" title="c10">x</font> be   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">d1</font>,<font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">M</font> #);<br/>

<b>reconsider </b><font color="Maroon" title="c11">y</font> = <font color="Maroon" title="c10">x</font> as   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="txt" href="clvect_1.html#E1:57"><i><font color="Green" title="E32">A1</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<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="c10">x</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="c11">y</font>
<b>by </b><i><a class="txt" href="clvect_1.html#E1:57"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="clvect_1.html#E4:57"><i><font color="Green" title="E35">A4</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon" title="c10">x</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>hence </b><a NAME="E8:57"/>
 <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c4">d1</font>,<font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">M</font> #) is    <a href="clvect_1.html#M1" title="CLVECT_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font>
 <b>by </b><i><a class="txt" href="clvect_1.html#E1:57"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="clvect_1.html#E3:57"><i><font color="Green" title="E34">A3</font></i></a>, <a class="txt" href="clvect_1.html#E4:57"><i><font color="Green" title="E35">A4</font></i></a>, <a class="txt" href="clvect_1.html#E5:57"><i><font color="Green" title="E36">A5</font></i></a>, <a class="ref" href="clvect_1.html#D5" target="_self" title="CLVECT_1:def.5">Def5</a></i>;<br/>


</div>
