<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">V</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">VectSp</a> of  <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> ;<br/><b>let </b><font color="Maroon" title="c2">l</font> be   <a href="hahnban1.html#NM2" title="HAHNBAN1:NM.2">linear-Functional</a> of <font color="Maroon" title="c1">V</font>;<br/>



<a NAME="E1:83"/><i><font color="Green" title="E42">A1</font></i>: 
 <a href="hahnban1.html#K12" title="HAHNBAN1:func.12">projIm</a> <font color="Maroon" title="c2">l</font> is <a href="hahnban.html#V2" title="HAHNBAN:attr.2">additive</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="hahnban1.html#K10" title="HAHNBAN1:func.10">RealVS</a> <font color="Maroon" title="c1">V</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="hahnban.html#D4" title="HAHNBAN:def.4">HAHNBAN:def 4</a><br/>

<a NAME="E1:83_1"/><i><font color="Green" title="E43">A2</font></i>: 
 <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">LoopStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>,the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c1">V</font>,the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">Zero</a> of <font color="Maroon" title="c1">V</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">LoopStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="hahnban1.html#K10" title="HAHNBAN1:func.10">RealVS</a> <font color="Maroon" title="c1">V</font></span>)</span>,the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <span class="p1">(<span class="default"><a href="hahnban1.html#K10" title="HAHNBAN1:func.10">RealVS</a> <font color="Maroon" title="c1">V</font></span>)</span>,the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">Zero</a> of <span class="p1">(<span class="default"><a href="hahnban1.html#K10" title="HAHNBAN1:func.10">RealVS</a> <font color="Maroon" title="c1">V</font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="hahnban1.html#D22" target="_self" title="HAHNBAN1:def.22">Def22</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">x1</font> = <font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c6">y1</font> = <font color="Maroon" title="c4">y</font> as   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Vector</a> of <font color="Maroon" title="c1">V</font> ;<br/>
<a NAME="E3:83_1"/>
<font color="Maroon" title="c3">x</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c6">y1</font>
 
<b>by </b><i><a class="txt" href="hahnban1.html#E1:83_1"><i><font color="Green" title="E43">A2</font></i></a></i>;<br/>
<b>hence </b><span class="p1">(<span class="default"><a href="hahnban1.html#K12" title="HAHNBAN1:func.12">projIm</a> <font color="Maroon" title="c2">l</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="c3">x</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c4">y</font></span>)</span> = 
 <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">x1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c6">y1</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="hahnban1.html#D24" target="_self" title="HAHNBAN1:def.24">Def24</a></i>
<br/>.= 
 <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x1</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">y1</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="hahnban1.html#D11" target="_self" title="HAHNBAN1:def.11">Def11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x1</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">y1</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="hahnban1.html#T16" target="_self" title="HAHNBAN1:th.16">Th16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x1</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hahnban1.html#K12" title="HAHNBAN1:func.12">projIm</a> <font color="Maroon" title="c2">l</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">y</font></span>)</span>
<b>by </b><i><a class="ref" href="hahnban1.html#D24" target="_self" title="HAHNBAN1:def.24">Def24</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hahnban1.html#K12" title="HAHNBAN1:func.12">projIm</a> <font color="Maroon" title="c2">l</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hahnban1.html#K12" title="HAHNBAN1:func.12">projIm</a> <font color="Maroon" title="c2">l</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">y</font></span>)</span>
<b>by </b><i><a class="ref" href="hahnban1.html#D24" target="_self" title="HAHNBAN1:def.24">Def24</a></i>
;<br/><br/>


</div><b>end;</b></div>
<a NAME="E2:83"/>
 <a href="hahnban1.html#K12" title="HAHNBAN1:func.12">projIm</a> <font color="Maroon" title="c2">l</font> is <a href="hahnban.html#V3" title="HAHNBAN:attr.3">homogeneous</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">x</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="hahnban1.html#K10" title="HAHNBAN1:func.10">RealVS</a> <font color="Maroon" title="c1">V</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="hahnban.html#D5" title="HAHNBAN:def.5">HAHNBAN:def 5</a><br/><b>let </b><font color="Maroon" title="c4">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>



<a NAME="E1:83_2"/>
 <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">LoopStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>,the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c1">V</font>,the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">Zero</a> of <font color="Maroon" title="c1">V</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">LoopStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="hahnban1.html#K10" title="HAHNBAN1:func.10">RealVS</a> <font color="Maroon" title="c1">V</font></span>)</span>,the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <span class="p1">(<span class="default"><a href="hahnban1.html#K10" title="HAHNBAN1:func.10">RealVS</a> <font color="Maroon" title="c1">V</font></span>)</span>,the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">Zero</a> of <span class="p1">(<span class="default"><a href="hahnban1.html#K10" title="HAHNBAN1:func.10">RealVS</a> <font color="Maroon" title="c1">V</font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="hahnban1.html#D22" target="_self" title="HAHNBAN1:def.22">Def22</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">x1</font> = <font color="Maroon" title="c3">x</font> as   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Vector</a> of <font color="Maroon" title="c1">V</font> ;<br/>
<a NAME="E3:83_2"/>
<font color="Maroon" title="c4">r</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><font color="Maroon" title="c4">r</font>,0</span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c5">x1</font>
 
<b>by </b><i><a class="ref" href="hahnban1.html#D22" target="_self" title="HAHNBAN1:def.22">Def22</a></i>;<br/>
<b>hence </b><span class="p1">(<span class="default"><a href="hahnban1.html#K12" title="HAHNBAN1:func.12">projIm</a> <font color="Maroon" title="c2">l</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="c4">r</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <font color="Maroon" title="c3">x</font></span>)</span> = 
 <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><span class="p3"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><font color="Maroon" title="c4">r</font>,0</span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c5">x1</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="hahnban1.html#D24" target="_self" title="HAHNBAN1:def.24">Def24</a></i>
<br/>.= 
 <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p1">(<span class="default"><span class="p2"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><font color="Maroon" title="c4">r</font>,0</span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x1</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="hahnban1.html#D12" target="_self" title="HAHNBAN1:def.12">Def12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p3"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><font color="Maroon" title="c4">r</font>,0</span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x1</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x1</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p3"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><font color="Maroon" title="c4">r</font>,0</span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="hahnban1.html#T17" target="_self" title="HAHNBAN1:th.17">Th17</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p3"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><font color="Maroon" title="c4">r</font>,0</span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x1</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x1</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> 0</span>)</span>
<b>by </b><i><a class="txt" href="hahnban1.html#E18"><i><font color="Green" title="E9">Lm3</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c4">r</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x1</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="hahnban1.html#E18"><i><font color="Green" title="E9">Lm3</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c4">r</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hahnban1.html#K12" title="HAHNBAN1:func.12">projIm</a> <font color="Maroon" title="c2">l</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font></span>)</span>
<b>by </b><i><a class="ref" href="hahnban1.html#D24" target="_self" title="HAHNBAN1:def.24">Def24</a></i>
;<br/><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:83"/>
 <a href="hahnban1.html#K12" title="HAHNBAN1:func.12">projIm</a> <font color="Maroon" title="c2">l</font> is   <a href="hahnban.html#NM3" title="HAHNBAN:NM.3">linear-Functional</a> of <span class="p1">(<span class="default"><a href="hahnban1.html#K10" title="HAHNBAN1:func.10">RealVS</a> <font color="Maroon" title="c1">V</font></span>)</span>
 <b>by </b><i><a class="txt" href="hahnban1.html#E1:83"><i><font color="Green" title="E42">A1</font></i></a></i>;<br/>


</div>
