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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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">V1</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font>  st <font color="Olive" title="b1">V1</font> is  <a href="clvect_1.html#V6" title="CLVECT_1:attr.6">linearly-closed</a>  &amp;  not <font color="Olive" title="b1">V1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  holds <br/> <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Olive" title="b1">V1</font>,<span class="p1">(<span class="default"><a href="csspace.html#K8" title="CSSPACE:func.8">Zero_</a> (<font color="Olive" title="b1">V1</font>,<font color="Maroon" title="c1">V</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="csspace.html#K6" title="CSSPACE:func.6">Add_</a> (<font color="Olive" title="b1">V1</font>,<font color="Maroon" title="c1">V</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="csspace.html#K7" title="CSSPACE:func.7">Mult_</a> (<font color="Olive" title="b1">V1</font>,<font color="Maroon" title="c1">V</font>)</span>)</span> #) is    <a href="clvect_1.html#M1" title="CLVECT_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font></span><br/><span class="kw">let </span><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>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">V1</font> is  <a href="clvect_1.html#V6" title="CLVECT_1:attr.6">linearly-closed</a>  &amp;  not <font color="Maroon" title="c2">V1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c2">V1</font>,<span class="p1">(<span class="default"><a href="csspace.html#K8" title="CSSPACE:func.8">Zero_</a> (<font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c1">V</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="csspace.html#K6" title="CSSPACE:func.6">Add_</a> (<font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c1">V</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="csspace.html#K7" title="CSSPACE:func.7">Mult_</a> (<font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c1">V</font>)</span>)</span> #) is    <a href="clvect_1.html#M1" title="CLVECT_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:23"/><span class="lab"><font color="Green" title="E10">A1</font></span>: 
<font color="Maroon" title="c2">V1</font> is  <a href="clvect_1.html#V6" title="CLVECT_1:attr.6">linearly-closed</a> 
 <span class="kw">and </span><br/><a NAME="E2:23"/><span class="lab"><font color="Green" title="E11">A2</font></span>: 
 not <font color="Maroon" title="c2">V1</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c2">V1</font>,<span class="p1">(<span class="default"><a href="csspace.html#K8" title="CSSPACE:func.8">Zero_</a> (<font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c1">V</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="csspace.html#K6" title="CSSPACE:func.6">Add_</a> (<font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c1">V</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="csspace.html#K7" title="CSSPACE:func.7">Mult_</a> (<font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c1">V</font>)</span>)</span> #) is    <a href="clvect_1.html#M1" title="CLVECT_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font></span><br/>

<a NAME="E3:23"/><span class="lab"><font color="Green" title="E12">A3</font></span>: 
 <a href="csspace.html#K6" title="CSSPACE:func.6">Add_</a> (<font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c1">V</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="ref" href="csspace.html#D8" target="_self" onmouseover="rs('csspace/D8')" onmouseout="rh()">Def6</a></span>;<br/>
<a NAME="E4:23"/><span class="lab"><font color="Green" title="E13">A4</font></span>: 
 <a href="csspace.html#K7" title="CSSPACE:func.7">Mult_</a> (<font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c1">V</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="relat_1.html#K5" title="RELAT_1:func.5">|</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="ref" href="csspace.html#D9" target="_self" onmouseover="rs('csspace/D9')" onmouseout="rh()">Def7</a></span>;<br/>
<a NAME="E5:23"/>
 <a href="csspace.html#K8" title="CSSPACE:func.8">Zero_</a> (<font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c1">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="txt" href="#E2:23"><span class="lab"><font color="Green" title="E11">A2</font></span></a>, <a class="ref" href="csspace.html#D10" target="_self" onmouseover="rs('csspace/D10')" onmouseout="rh()">Def8</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:23"/>
 <a href="clvect_1.html#G1" title="CLVECT_1:aggr.1">CLSStruct</a>(# <font color="Maroon" title="c2">V1</font>,<span class="p1">(<span class="default"><a href="csspace.html#K8" title="CSSPACE:func.8">Zero_</a> (<font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c1">V</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="csspace.html#K6" title="CSSPACE:func.6">Add_</a> (<font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c1">V</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="csspace.html#K7" title="CSSPACE:func.7">Mult_</a> (<font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c1">V</font>)</span>)</span> #) is    <a href="clvect_1.html#M1" title="CLVECT_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:23"><span class="lab"><font color="Green" title="E11">A2</font></span></a>, <a class="txt" href="#E3:23"><span class="lab"><font color="Green" title="E12">A3</font></span></a>, <a class="txt" href="#E4:23"><span class="lab"><font color="Green" title="E13">A4</font></span></a>, <a class="ref" href="clvect_1.html#T43" onmouseover="rs('clvect_1/T43')" onmouseout="rh()">CLVECT_1:43</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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