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

<b>let </b><font color="Maroon">X</font> be   <a href="clvect_1.html#NM2">ComplexLinearSpace</a>, <font color="Maroon">Y</font> be   <a href="clvect_1.html#NM2">ComplexLinearSpace</a>;<br/>

<b>set </b><font color="Maroon">W</font> =  <a href="clopban1.html#K5" target="_self">LinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font>;<br/>
<a NAME="E1:27"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">v</font>, <font color="Olive">u</font> being  <a href="clvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="clopban1.html#K3" target="_self">ComplexVectSpace</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span>  st <font color="Olive">v</font> <a href="hidden.html#R2">in</a>  <a href="clopban1.html#K5" target="_self">LinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font> &amp; <font color="Olive">u</font> <a href="hidden.html#R2">in</a>  <a href="clopban1.html#K5" target="_self">LinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font> holds <br/><font color="Olive">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">u</font> <a href="hidden.html#R2">in</a>  <a href="clopban1.html#K5" target="_self">LinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">v</font> be   <a href="clvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="clopban1.html#K3" target="_self">ComplexVectSpace</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span>, <font color="Maroon">u</font> be   <a href="clvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="clopban1.html#K3" target="_self">ComplexVectSpace</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:27_1"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="clopban1.html#K5" target="_self">LinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="clopban1.html#K5" target="_self">LinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font> )
 ;<br/>

<a NAME="E2:27_1"/>
<font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">u</font> is   <a href="clopban1.html#NM1">LinearOperator</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">u</font> as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font> <b>by </b><i><a class="ref" href="funct_2.html#T121">FUNCT_2:121</a></i>;<br/>
<a NAME="E2:27_1_1"/>
<font color="Maroon">f</font> is   <a href="clopban1.html#NM1">LinearOperator</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:27_1_1_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">f</font> is <a href="clopban1.html#V1" target="_self">additive</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font>, <font color="Maroon">y</font> be   <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="clopban1.html#D3" target="_self">CLOPBAN1:def 3</a><br/>

<b>reconsider </b><font color="Maroon">v'</font> = <font color="Maroon">v</font>, <font color="Maroon">u'</font> = <font color="Maroon">u</font> as    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font> ;<br/>
<a NAME="E2:27_1_1_1_1"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="lopban_1.html#K2">FuncAdd</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <a href="funcsdom.html#K1">.</a> <font color="Maroon">u'</font>,<font color="Maroon">v'</font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D3">RLVECT_1:def 3</a></i>;<br/>
<a NAME="E3:27_1_1_1_1"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">u'</font> is   <a href="clopban1.html#NM1">LinearOperator</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:27_1_1_1_1"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">v'</font> is   <a href="clopban1.html#NM1">LinearOperator</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font>
 
<b>by </b><i>, </i>;<br/>
<b>thus </b><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">u'</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">v'</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="lopban_1.html#T3">LOPBAN_1:3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">u'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">u'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">v'</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">u'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">u'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">v'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">u'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><font color="Maroon">u'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">v'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">v'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">u'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><font color="Maroon">v'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">u'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">v'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">u'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">v'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i>, <a class="ref" href="lopban_1.html#T3">LOPBAN_1:3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">u'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">v'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i>, <a class="ref" href="lopban_1.html#T3">LOPBAN_1:3</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E2:27_1_1_1"/>
<font color="Maroon">f</font> is <a href="clopban1.html#V2" target="_self">homogeneous</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="clopban1.html#D4" target="_self">CLOPBAN1:def 4</a><br/><b>let </b><font color="Maroon">s</font> be   <a href="cfunct_1.html#NM1">Complex</a>;<br/>



<b>reconsider </b><font color="Maroon">v'</font> = <font color="Maroon">v</font>, <font color="Maroon">u'</font> = <font color="Maroon">u</font> as    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font> ;<br/>
<a NAME="E2:27_1_1_1_2"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="lopban_1.html#K2">FuncAdd</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <a href="funcsdom.html#K1">.</a> <font color="Maroon">u'</font>,<font color="Maroon">v'</font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D3">RLVECT_1:def 3</a></i>;<br/>
<a NAME="E3:27_1_1_1_2"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">u'</font> is   <a href="clopban1.html#NM1">LinearOperator</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:27_1_1_1_2"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">v'</font> is   <a href="clopban1.html#NM1">LinearOperator</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font>
 
<b>by </b><i>, </i>;<br/>
<b>thus </b><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">x</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">u'</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">v'</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="lopban_1.html#T3">LOPBAN_1:3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="clvect_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">u'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">v'</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="clvect_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">u'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="clvect_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">v'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">s</font> <a href="clvect_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">u'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">v'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="clvect_1.html#D2">CLVECT_1:def 2</a></i>
<br/>.= 
<font color="Maroon">s</font> <a href="clvect_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, <a class="ref" href="lopban_1.html#T3">LOPBAN_1:3</a></i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:27_1_1_1"/>
<font color="Maroon">f</font> is   <a href="clopban1.html#NM1">LinearOperator</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:27_1_1"/>
<font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">u</font> is   <a href="clopban1.html#NM1">LinearOperator</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:27_1"/>
<font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="clopban1.html#K5" target="_self">LinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E2:27"/>
for <font color="Olive">c</font> being  <a href="cfunct_1.html#NM1">Complex</a><br/> for <font color="Olive">v</font> being  <a href="clvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="clopban1.html#K3" target="_self">ComplexVectSpace</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span>  st <font color="Olive">v</font> <a href="hidden.html#R2">in</a>  <a href="clopban1.html#K5" target="_self">LinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font> holds <br/><font color="Olive">c</font> <a href="clvect_1.html#K1">*</a> <font color="Olive">v</font> <a href="hidden.html#R2">in</a>  <a href="clopban1.html#K5" target="_self">LinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c</font> be   <a href="cfunct_1.html#NM1">Complex</a>;<br/><b>let </b><font color="Maroon">v</font> be   <a href="clvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="clopban1.html#K3" target="_self">ComplexVectSpace</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span>;<br/>



<b>assume </b><a NAME="E1:27_2"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="clopban1.html#K5" target="_self">LinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font>
 ;<br/>

<a NAME="E2:27_2"/>
<font color="Maroon">c</font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">v</font> is   <a href="clopban1.html#NM1">LinearOperator</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">c</font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">v</font> as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font> <b>by </b><i><a class="ref" href="funct_2.html#T121">FUNCT_2:121</a></i>;<br/>
<a NAME="E2:27_2_1"/>
<font color="Maroon">f</font> is   <a href="clopban1.html#NM1">LinearOperator</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:27_2_1_1"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">f</font> is <a href="clopban1.html#V1" target="_self">additive</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font>, <font color="Maroon">y</font> be   <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="clopban1.html#D3" target="_self">CLOPBAN1:def 3</a><br/>

<b>reconsider </b><font color="Maroon">v'</font> = <font color="Maroon">v</font> as    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font> ;<br/>
<a NAME="E2:27_2_1_1_1"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="clopban1.html#K2" target="_self">FuncExtMult</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">v'</font></span><a href="domain_1.html#K1">]</a></span>
 
<b>by </b><i><a class="ref" href="clvect_1.html#D1">CLVECT_1:def 1</a></i>;<br/>
<a NAME="E3:27_2_1_1_1"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">v'</font> is   <a href="clopban1.html#NM1">LinearOperator</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font>
 
<b>by </b><i>, </i>;<br/>
<b>thus </b><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font></span>)</span> = 
<font color="Maroon">c</font> <a href="clvect_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">v'</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">c</font> <a href="clvect_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">v'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="clvect_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">v'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="clvect_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">v'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="clvect_1.html#D2">CLVECT_1:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="clvect_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">v'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i>, </i>
;<br/>


</div><b>end;</b></div>
<a NAME="E2:27_2_1_1"/>
<font color="Maroon">f</font> is <a href="clopban1.html#V2" target="_self">homogeneous</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="clopban1.html#D4" target="_self">CLOPBAN1:def 4</a><br/><b>let </b><font color="Maroon">s</font> be   <a href="cfunct_1.html#NM1">Complex</a>;<br/>



<b>reconsider </b><font color="Maroon">v'</font> = <font color="Maroon">v</font> as    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font> ;<br/>
<a NAME="E2:27_2_1_1_2"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="clopban1.html#K2" target="_self">FuncExtMult</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">v'</font></span><a href="domain_1.html#K1">]</a></span>
 
<b>by </b><i><a class="ref" href="clvect_1.html#D1">CLVECT_1:def 1</a></i>;<br/>
<a NAME="E3:27_2_1_1_2"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">v'</font> is   <a href="clopban1.html#NM1">LinearOperator</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font>
 
<b>by </b><i>, </i>;<br/>
<b>thus </b><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">x</font></span>)</span> = 
<font color="Maroon">c</font> <a href="clvect_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">v'</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">c</font> <a href="clvect_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="clvect_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">v'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="complex1.html#K9">*</a> <font color="Maroon">s</font></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">v'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="clvect_1.html#D2">CLVECT_1:def 2</a></i>
<br/>.= 
<font color="Maroon">s</font> <a href="clvect_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="clvect_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">v'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="clvect_1.html#D2">CLVECT_1:def 2</a></i>
<br/>.= 
<font color="Maroon">s</font> <a href="clvect_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, </i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:27_2_1_1"/>
<font color="Maroon">f</font> is   <a href="clopban1.html#NM1">LinearOperator</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:27_2_1"/>
<font color="Maroon">c</font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">v</font> is   <a href="clopban1.html#NM1">LinearOperator</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:27_2"/>
<font color="Maroon">c</font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="clopban1.html#K5" target="_self">LinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:27"/>
 <a href="clopban1.html#K5" target="_self">LinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font> is <a href="clvect_1.html#V3">lineary-closed</a>
 <b>by </b><i>, <a class="ref" href="clvect_1.html#D3">CLVECT_1:def 3</a></i>;<br/>


</div>
