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<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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">z</font> being  <a href="cfunct_1.html#NM1" title="CFUNCT_1:NM.1">Complex</a><br/> for <font color="Olive" title="b3">W</font> being   <a href="clvect_1.html#M1" title="CLVECT_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font>  st <font color="Olive" title="b2">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a>  &amp; <font color="Olive" title="b2">z</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">v</font> <a href="clvect_1.html#K5" title="CLVECT_1:func.5">+</a> <font color="Olive" title="b3">W</font> holds <br/><font color="Olive" title="b1">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Olive" title="b3">W</font></span><br/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> 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">W</font> being   <a href="clvect_1.html#M1" title="CLVECT_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font>  st <font color="Olive" title="b1">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a>  &amp; <font color="Olive" title="b1">z</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">v</font> <a href="clvect_1.html#K5" title="CLVECT_1:func.5">+</a> <font color="Olive" title="b2">W</font> holds <br/><font color="Maroon" title="c2">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Olive" title="b2">W</font></span><br/><b>let </b><font color="Maroon" title="c3">z</font> be   <a href="cfunct_1.html#NM1" title="CFUNCT_1:NM.1">Complex</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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>  st <font color="Maroon" title="c3">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a>  &amp; <font color="Maroon" title="c3">z</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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> holds <br/><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/><b>let </b><font color="Maroon" title="c4">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a>  &amp; <font color="Maroon" title="c3">z</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">v</font> <a href="clvect_1.html#K5" title="CLVECT_1:func.5">+</a> <font color="Maroon" title="c4">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="c4">W</font> )</span><br/>







<b>assume </b><b>that </b><br/><a NAME="E1:92"/><i><font color="Green" title="E58">A1</font></i>: 
<font color="Maroon" title="c3">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a> 
 <b>and </b><br/><a NAME="E2:92"/><i><font color="Green" title="E59">A2</font></i>: 
<font color="Maroon" title="c3">z</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">v</font> <a href="clvect_1.html#K5" title="CLVECT_1:func.5">+</a> <font color="Maroon" title="c4">W</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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="c4">W</font></span><br/>

<b>consider </b><font color="Maroon" title="c5">u</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="E4:92"/><i><font color="Green" title="E60">A3</font></i>: 
<font color="Maroon" title="c3">z</font> <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> <font color="Maroon" title="c2">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c5">u</font>
 <b>and </b><br/><a NAME="E5:92"/><i><font color="Green" title="E61">A4</font></i>: 
<font color="Maroon" title="c5">u</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">W</font>
 <b>by </b><i><a class="txt" href="clvect_1.html#E2:92"><i><font color="Green" title="E59">A2</font></i></a></i>;<br/>
<font color="Maroon" title="c5">u</font> = 
<font color="Maroon" title="c5">u</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="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></i>
<br/>.= 
<font color="Maroon" title="c5">u</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <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> <font color="Maroon" title="c2">v</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T28" title="RLVECT_1:th.28">RLVECT_1:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">z</font> <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="c2">v</font>
<b>by </b><i><a class="txt" href="clvect_1.html#E4:92"><i><font color="Green" title="E60">A3</font></i></a>, <a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">z</font> <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> <span class="p1">(<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>
<b>by </b><i><a class="ref" href="clvect_1.html#D2" target="_self" title="CLVECT_1:def.2">Def2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</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>
<b>by </b><i><a class="ref" href="clvect_1.html#T11" target="_self" title="CLVECT_1:th.11">Th11</a></i>
;<br/>
<a NAME="E7:92"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</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="c5">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</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"><font color="Maroon" title="c3">z</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</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> &amp; <font color="Maroon" title="c3">z</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<b>by </b><i><a class="txt" href="clvect_1.html#E1:92"><i><font color="Green" title="E58">A1</font></i></a>, <a class="ref" href="clvect_1.html#D2" target="_self" title="CLVECT_1:def.2">Def2</a></i>;<br/>
<a NAME="E8:92"/><b>then </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="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"><font color="Maroon" title="c3">z</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</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="c5">u</font>
 
<b>by </b><i><a class="ref" href="complex1.html#D7" title="COMPLEX1:def.7">COMPLEX1:def 7</a>, <a class="ref" href="xcmplx_0.html#D7" title="XCMPLX_0:def.7">XCMPLX_0:def 7</a></i>;<br/>
<a NAME="E9:92"/><b>then </b>
<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"><font color="Maroon" title="c3">z</font> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</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="c5">u</font>
 
<b>by </b><i><a class="ref" href="clvect_1.html#D2" target="_self" title="CLVECT_1:def.2">Def2</a></i>;<br/>
<b>hence </b><a NAME="E10:92"/>
<font color="Maroon" title="c2">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">W</font>
 <b>by </b><i><a class="txt" href="clvect_1.html#E5:92"><i><font color="Green" title="E61">A4</font></i></a>, <a class="ref" href="clvect_1.html#T41" target="_self" title="CLVECT_1:th.41">Th41</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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