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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">u</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><br/> for <font color="Olive" title="b3">C</font> being   <a href="clvect_1.html#M2" title="CLVECT_1:mode.2">Coset</a> of <font color="Olive" title="b2">W</font> holds <br/> ( <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">C</font> iff <font color="Olive" title="b3">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">u</font> <a href="clvect_1.html#K5" title="CLVECT_1:func.5">+</a> <font color="Olive" title="b2">W</font> )</span><br/><b>let </b><font color="Maroon" title="c2">u</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">W</font> being   <a href="clvect_1.html#M1" title="CLVECT_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font><br/> for <font color="Olive" title="b2">C</font> being   <a href="clvect_1.html#M2" title="CLVECT_1:mode.2">Coset</a> of <font color="Olive" title="b1">W</font> holds <br/> ( <font color="Maroon" title="c2">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">C</font> iff <font color="Olive" title="b2">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">u</font> <a href="clvect_1.html#K5" title="CLVECT_1:func.5">+</a> <font color="Olive" title="b1">W</font> )</span><br/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">C</font> being   <a href="clvect_1.html#M2" title="CLVECT_1:mode.2">Coset</a> of <font color="Maroon" title="c3">W</font> holds <br/> ( <font color="Maroon" title="c2">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> iff <font color="Olive" title="b1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">u</font> <a href="clvect_1.html#K5" title="CLVECT_1:func.5">+</a> <font color="Maroon" title="c3">W</font> )</span><br/><b>let </b><font color="Maroon" title="c4">C</font> be    <a href="clvect_1.html#M2" title="CLVECT_1:mode.2">Coset</a> of <font color="Maroon" title="c3">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">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> iff <font color="Maroon" title="c4">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">u</font> <a href="clvect_1.html#K5" title="CLVECT_1:func.5">+</a> <font color="Maroon" title="c3">W</font> )</span><br/>







<b>thus </b><a NAME="E1:112"/>
( <font color="Maroon" title="c2">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> implies <font color="Maroon" title="c4">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">u</font> <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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">u</font> <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">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="112_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:112_1"/><i><font color="Green" title="E67">A1</font></i>: 
<font color="Maroon" title="c2">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">u</font> <a href="clvect_1.html#K5" title="CLVECT_1:func.5">+</a> <font color="Maroon" title="c3">W</font></span><br/>

<a NAME="E2:112_1"/>
 ex <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> st <font color="Maroon" title="c4">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v</font> <a href="clvect_1.html#K5" title="CLVECT_1:func.5">+</a> <font color="Maroon" title="c3">W</font>
 
<b>by </b><i><a class="ref" href="clvect_1.html#D9" target="_self" title="CLVECT_1:def.9">Def9</a></i>;<br/>
<b>hence </b><a NAME="E3:112_1"/>
<font color="Maroon" title="c4">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">u</font> <a href="clvect_1.html#K5" title="CLVECT_1:func.5">+</a> <font color="Maroon" title="c3">W</font>
 <b>by </b><i><a class="txt" href="clvect_1.html#E1:112_1"><i><font color="Green" title="E67">A1</font></i></a>, <a class="ref" href="clvect_1.html#T74" target="_self" title="CLVECT_1:th.74">Th74</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>thus </b><a NAME="E2:112"/>
( <font color="Maroon" title="c4">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">u</font> <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">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> )
 <b>by </b><i><a class="ref" href="clvect_1.html#T63" target="_self" title="CLVECT_1:th.63">Th63</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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