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<b>let </b><font color="Maroon" title="c1">V</font> be   <a href="vectsp_1.html#NM7" title="VECTSP_1:NM.7">VectSp</a> of  <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> ;<br/><b>let </b><font color="Maroon" title="c2">W</font> be    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font>;<br/><b>let </b><font color="Maroon" title="c3">f</font> be   <a href="hermitan.html#NM1" title="HERMITAN:NM.1">antilinear-Functional</a> of <font color="Maroon" title="c1">V</font>;<br/>





<b>assume </b><a NAME="E1:38"/><i><font color="Green" title="E20">A1</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="vectsp10.html#K8" title="VECTSP10:func.8">ker</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hermitan.html#K1" title="HERMITAN:func.1">*'</a> </span>)</span>
 ;<br/>

<a NAME="E2:38"/><i><font color="Green" title="E21">A2</font></i>: 
 <a href="vectsp10.html#K8" title="VECTSP10:func.8">ker</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hermitan.html#K1" title="HERMITAN:func.1">*'</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp10.html#K8" title="VECTSP10:func.8">ker</a> <font color="Maroon" title="c3">f</font>
 
<b>by </b><i><a class="ref" href="hermitan.html#T26" target="_self" title="HERMITAN:th.26">Th26</a></i>;<br/>
<b>set </b><font color="Maroon" title="c4">qf</font> =  <a href="vectsp10.html#K10" title="VECTSP10:func.10">QFunctional</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">W</font>;<br/>
<b>set </b><font color="Maroon" title="c5">vq</font> =  <a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">W</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">A</font> be   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">W</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c7">r</font> be   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Scalar</a> of ;<br/><b>set </b><font color="Maroon" title="c8">C</font> =  <a href="vectsp10.html#K2" title="VECTSP10:func.2">CosetSet</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">W</font>;<br/><a NAME="E1:38_1"/><i><font color="Green" title="E22">A3</font></i>: 
 <a href="vectsp10.html#K2" title="VECTSP10:func.2">CosetSet</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">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 <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">W</font></span>)</span>
 <b>by </b><i><a class="ref" href="vectsp10.html#D6" title="VECTSP10:def.6">VECTSP10:def 6</a></i>;<br/><a NAME="E2:38_1"/><b>then </b>
<font color="Maroon" title="c6">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="vectsp10.html#K2" title="VECTSP10:func.2">CosetSet</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">W</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c9">aa</font> being    <a href="vectsp_4.html#M2" title="VECTSP_4:mode.2">Coset</a> of <font color="Maroon" title="c2">W</font><b> such that </b><br/><a NAME="E4:38_1"/><i><font color="Green" title="E23">A4</font></i>: 
<font color="Maroon" title="c6">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">aa</font>
 ;<br/><b>consider </b><font color="Maroon" title="c10">a</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E6:38_1"/><i><font color="Green" title="E24">A5</font></i>: 
<font color="Maroon" title="c9">aa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">a</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <font color="Maroon" title="c2">W</font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#D6" title="VECTSP_4:def.6">VECTSP_4:def 6</a></i>;<br/><font color="Maroon" title="c7">r</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c6">A</font> = 
<span class="p1">(<span class="default"><a href="vectsp10.html#K5" title="VECTSP10:func.5">lmultCoset</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">W</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">r</font>,<font color="Maroon" title="c6">A</font>
<b>by </b><i><a class="ref" href="vectsp10.html#D6" title="VECTSP10:def.6">VECTSP10:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c10">a</font></span>)</span> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <font color="Maroon" title="c2">W</font>
<b>by </b><i><a class="txt" href="hermitan.html#E1:38_1"><i><font color="Green" title="E22">A3</font></i></a>, <a class="txt" href="hermitan.html#E4:38_1"><i><font color="Green" title="E23">A4</font></i></a>, <a class="txt" href="hermitan.html#E6:38_1"><i><font color="Green" title="E24">A5</font></i></a>, <a class="ref" href="vectsp10.html#D5" title="VECTSP10:def.5">VECTSP10:def 5</a></i>
;<br/><b>hence </b><span class="p1">(<span class="default"><a href="vectsp10.html#K10" title="VECTSP10:func.10">QFunctional</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">W</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c6">A</font></span>)</span> = 
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c10">a</font></span>)</span>
<b>by </b><i><a class="txt" href="hermitan.html#E1:38"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="hermitan.html#E2:38"><i><font color="Green" title="E21">A2</font></i></a>, <a class="ref" href="vectsp10.html#D12" title="VECTSP10:def.12">VECTSP10:def 12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="complfld.html#K2" title="COMPLFLD:func.2">*'</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">a</font></span>)</span>
<b>by </b><i><a class="ref" href="hermitan.html#D1" target="_self" title="HERMITAN:def.1">Def1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="complfld.html#K2" title="COMPLFLD:func.2">*'</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vectsp10.html#K10" title="VECTSP10:func.10">QFunctional</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">W</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">A</font></span>)</span>
<b>by </b><i><a class="txt" href="hermitan.html#E1:38"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="hermitan.html#E2:38"><i><font color="Green" title="E21">A2</font></i></a>, <a class="txt" href="hermitan.html#E4:38_1"><i><font color="Green" title="E23">A4</font></i></a>, <a class="txt" href="hermitan.html#E6:38_1"><i><font color="Green" title="E24">A5</font></i></a>, <a class="ref" href="vectsp10.html#D12" title="VECTSP10:def.12">VECTSP10:def 12</a></i>
;<br/><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:38"/>
 <a href="vectsp10.html#K10" title="VECTSP10:func.10">QFunctional</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">W</font> is <a href="hermitan.html#V1" title="HERMITAN:attr.1">cmplxhomogeneous</a>
 <b>by </b><i><a class="ref" href="hermitan.html#D1" target="_self" title="HERMITAN:def.1">Def1</a></i>;<br/>


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