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<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>, <font color="Maroon">Z</font> be   <a href="clvect_1.html#NM2">ComplexLinearSpace</a>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="clopban1.html#NM1">LinearOperator</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font>;<br/><b>let </b><font color="Maroon">g</font> be   <a href="clopban1.html#NM1">LinearOperator</a> of <font color="Maroon">Y</font>,<font color="Maroon">Z</font>;<br/>





<div><i><font color="Green">E42</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">v</font> be   <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font>, <font color="Maroon">w</font> be   <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">w</font></span>)</span> = 
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">w</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <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">v</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="clopban1.html#D3">CLOPBAN1:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="clopban1.html#D3">CLOPBAN1:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</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">g</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">v</font> be   <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">z</font> be   <a href="cfunct_1.html#NM1">Complex</a>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">v</font></span>)</span> = 
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">v</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="clvect_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="clopban1.html#D4">CLOPBAN1:def 4</a></i>
<br/>.= 
<font color="Maroon">z</font> <a href="clvect_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="clopban1.html#D4">CLOPBAN1:def 4</a></i>
<br/>.= 
<font color="Maroon">z</font> <a href="clvect_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:1"/>
<font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font> is   <a href="clopban1.html#NM1">LinearOperator</a> of <font color="Maroon">X</font>,<font color="Maroon">Z</font>
 <b>by </b><i>, <a class="ref" href="clopban1.html#D3">CLOPBAN1:def 3</a>, <a class="ref" href="clopban1.html#D4">CLOPBAN1:def 4</a></i>;<br/>


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