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

<span class="kw">let </span><font color="Maroon" title="c1">V</font> be   <a href="clvect_1.html#NM2" title="CLVECT_1:NM.2">ComplexLinearSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">v</font> being  <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font><br/> for <font color="Olive" title="b2">W</font> being   <a href="clvect_1.html#M1" title="CLVECT_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font> holds <br/> ( <font color="Olive" title="b1">v</font> <a href="clvect_1.html#K5" title="CLVECT_1:func.5">+</a> <font color="Olive" title="b2">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Olive" title="b1">v</font></span>)</span> <a href="clvect_1.html#K5" title="CLVECT_1:func.5">+</a> <font color="Olive" title="b2">W</font> iff <font color="Olive" title="b1">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Olive" title="b2">W</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">v</font> be   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">W</font> being   <a href="clvect_1.html#M1" title="CLVECT_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font> holds <br/> ( <font color="Maroon" title="c2">v</font> <a href="clvect_1.html#K5" title="CLVECT_1:func.5">+</a> <font color="Olive" title="b1">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c2">v</font></span>)</span> <a href="clvect_1.html#K5" title="CLVECT_1:func.5">+</a> <font color="Olive" title="b1">W</font> iff <font color="Maroon" title="c2">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Olive" title="b1">W</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">W</font> be    <a href="clvect_1.html#M1" title="CLVECT_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">v</font> <a href="clvect_1.html#K5" title="CLVECT_1:func.5">+</a> <font color="Maroon" title="c3">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c2">v</font></span>)</span> <a href="clvect_1.html#K5" title="CLVECT_1:func.5">+</a> <font color="Maroon" title="c3">W</font> iff <font color="Maroon" title="c2">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">W</font> )</span><br/>





<span class="kw">thus </span><a NAME="E1:89"/>
( <font color="Maroon" title="c2">v</font> <a href="clvect_1.html#K5" title="CLVECT_1:func.5">+</a> <font color="Maroon" title="c3">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c2">v</font></span>)</span> <a href="clvect_1.html#K5" title="CLVECT_1:func.5">+</a> <font color="Maroon" title="c3">W</font> implies <font color="Maroon" title="c2">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">W</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">W</font> implies <font color="Maroon" title="c2">v</font> <a href="clvect_1.html#K5" title="CLVECT_1:func.5">+</a> <font color="Maroon" title="c3">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c2">v</font></span>)</span> <a href="clvect_1.html#K5" title="CLVECT_1:func.5">+</a> <font color="Maroon" title="c3">W</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:89_1"/>
<font color="Maroon" title="c2">v</font> <a href="clvect_1.html#K5" title="CLVECT_1:func.5">+</a> <font color="Maroon" title="c3">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c2">v</font></span>)</span> <a href="clvect_1.html#K5" title="CLVECT_1:func.5">+</a> <font color="Maroon" title="c3">W</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">W</font></span><br/>

<a NAME="E2:89_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c2">v</font></span>)</span> <a href="clvect_1.html#K5" title="CLVECT_1:func.5">+</a> <font color="Maroon" title="c3">W</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="clvect_1.html#T63" target="_self" title="CLVECT_1:th.63">Th63</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">u</font> being   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font><span class="kw"> such that </span><br/><a NAME="E4:89_1"/><span class="lab"><font color="Green" title="E58">A1</font></span>: 
<font color="Maroon" title="c2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c2">v</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">u</font>
 <span class="kw">and </span><br/><a NAME="E5:89_1"/><span class="lab"><font color="Green" title="E59">A2</font></span>: 
<font color="Maroon" title="c4">u</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">W</font>
 ;<br/>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font> = 
<font color="Maroon" title="c2">v</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c2">v</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">u</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:89_1"><span class="lab"><font color="Green" title="E58">A1</font></span></a>, <a class="ref" href="rlvect_1.html#T28" title="RLVECT_1:th.28">RLVECT_1:28</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">v</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c2">v</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">u</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T41" title="RLVECT_1:th.41">RLVECT_1:41</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c2">v</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">u</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T30" title="RLVECT_1:th.30">RLVECT_1:30</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><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="c2">v</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c2">v</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">u</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="clvect_1.html#D5" target="_self" title="CLVECT_1:def.5">Def5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><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="c2">v</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><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="c2">v</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">u</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="clvect_1.html#D5" target="_self" title="CLVECT_1:def.5">Def5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a>  <a href="complex1.html#K8" title="COMPLEX1:func.8">+</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="c2">v</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">u</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="clvect_1.html#D3" target="_self" title="CLVECT_1:def.3">Def3</a></span>
;<br/>
<a NAME="E7:89_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a>  <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a> </span>)</span> <a href="complex1.html#K12" title="COMPLEX1:func.12">"</a> </span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a>  <a href="complex1.html#K8" title="COMPLEX1:func.8">+</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="c2">v</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a>  <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a> </span>)</span> <a href="complex1.html#K12" title="COMPLEX1:func.12">"</a> </span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c4">u</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T35" title="RLVECT_1:th.35">RLVECT_1:35</a></span>;<br/>
<a NAME="E8:89_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a>  <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a> </span>)</span> <a href="complex1.html#K12" title="COMPLEX1:func.12">"</a> </span>)</span> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <span class="p2">(<span class="default"><a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a>  <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a> </span>)</span></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a>  <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a> </span>)</span> <a href="complex1.html#K12" title="COMPLEX1:func.12">"</a> </span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c4">u</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="clvect_1.html#D4" target="_self" title="CLVECT_1:def.4">Def4</a></span>;<br/>
<a NAME="E9:89_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a>  <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a> </span>)</span> <a href="complex1.html#K12" title="COMPLEX1:func.12">"</a> </span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c4">u</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="clvect_1.html#D5" target="_self" title="CLVECT_1:def.5">Def5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:89_1"/>
<font color="Maroon" title="c2">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:89_1"><span class="lab"><font color="Green" title="E59">A2</font></span></a>, <a class="ref" href="clvect_1.html#T41" target="_self" title="CLVECT_1:th.41">Th41</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E2:89"/><span class="lab"><font color="Green" title="E58">A3</font></span>: 
<font color="Maroon" title="c2">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">W</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">v</font> <a href="clvect_1.html#K5" title="CLVECT_1:func.5">+</a> <font color="Maroon" title="c3">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c2">v</font></span>)</span> <a href="clvect_1.html#K5" title="CLVECT_1:func.5">+</a> <font color="Maroon" title="c3">W</font></span><br/>

<a NAME="E3:89"/><span class="kw">then </span>
<font color="Maroon" title="c2">v</font> <a href="clvect_1.html#K5" title="CLVECT_1:func.5">+</a> <font color="Maroon" title="c3">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">W</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="clvect_1.html#E82"><span class="lab"><font color="Green" title="E49">Lm5</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:89"/>
<font color="Maroon" title="c2">v</font> <a href="clvect_1.html#K5" title="CLVECT_1:func.5">+</a> <font color="Maroon" title="c3">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c2">v</font></span>)</span> <a href="clvect_1.html#K5" title="CLVECT_1:func.5">+</a> <font color="Maroon" title="c3">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:89"><span class="lab"><font color="Green" title="E58">A3</font></span></a>, <a class="ref" href="clvect_1.html#T71" target="_self" title="CLVECT_1:th.71">Th71</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
