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<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">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="Olive" title="b2">C</font> is  <a href="clvect_1.html#V6" title="CLVECT_1:attr.6">linearly-closed</a>  iff <font color="Olive" title="b2">C</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="Olive" title="b1">W</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">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"> 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="c2">W</font> holds <br/> ( <font color="Olive" title="b1">C</font> is  <a href="clvect_1.html#V6" title="CLVECT_1:attr.6">linearly-closed</a>  iff <font color="Olive" title="b1">C</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="c2">W</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">C</font> be    <a href="clvect_1.html#M2" title="CLVECT_1:mode.2">Coset</a> of <font color="Maroon" title="c2">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="c3">C</font> is  <a href="clvect_1.html#V6" title="CLVECT_1:attr.6">linearly-closed</a>  iff <font color="Maroon" title="c3">C</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="c2">W</font> )</span><br/>





<span class="kw">thus </span><a NAME="E1:104"/>
( <font color="Maroon" title="c3">C</font> is  <a href="clvect_1.html#V6" title="CLVECT_1:attr.6">linearly-closed</a>  implies <font color="Maroon" title="c3">C</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="c2">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="c3">C</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="c2">W</font> implies <font color="Maroon" title="c3">C</font> is  <a href="clvect_1.html#V6" title="CLVECT_1:attr.6">linearly-closed</a>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="104_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:104_1"/><span class="lab"><font color="Green" title="E68">A1</font></span>: 
<font color="Maroon" title="c3">C</font> is  <a href="clvect_1.html#V6" title="CLVECT_1:attr.6">linearly-closed</a> 
 ; <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="c3">C</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="c2">W</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c4">v</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="E3:104_1"/><span class="lab"><font color="Green" title="E69">A2</font></span>: 
<font color="Maroon" title="c3">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v</font> <a href="clvect_1.html#K5" title="CLVECT_1:func.5">+</a> <font color="Maroon" title="c2">W</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="clvect_1.html#D12" target="_self" title="CLVECT_1:def.12">Def12</a></span>;<br/>
<a NAME="E4:104_1"/>
<font color="Maroon" title="c3">C</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:104_1"><span class="lab"><font color="Green" title="E69">A2</font></span></a>, <a class="ref" href="clvect_1.html#T63" target="_self" title="CLVECT_1:th.63">Th63</a></span>;<br/>
<a NAME="E5:104_1"/><span class="kw">then </span>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">v</font> <a href="clvect_1.html#K5" title="CLVECT_1:func.5">+</a> <font color="Maroon" title="c2">W</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:104_1"><span class="lab"><font color="Green" title="E68">A1</font></span></a>, <a class="txt" href="#E3:104_1"><span class="lab"><font color="Green" title="E69">A2</font></span></a>, <a class="ref" href="clvect_1.html#T21" target="_self" title="CLVECT_1:th.21">Th21</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:104_1"/>
<font color="Maroon" title="c3">C</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="c2">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:104_1"><span class="lab"><font color="Green" title="E69">A2</font></span></a>, <a class="ref" href="clvect_1.html#T67" target="_self" title="CLVECT_1:th.67">Th67</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">thus </span><a NAME="E2:104"/>
( <font color="Maroon" title="c3">C</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="c2">W</font> implies <font color="Maroon" title="c3">C</font> is  <a href="clvect_1.html#V6" title="CLVECT_1:attr.6">linearly-closed</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="clvect_1.html#E47"><span class="lab"><font color="Green" title="E27">Lm3</font></span></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/>


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