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<b>let </b><font color="Maroon">X</font> be   <a href="clvect_1.html#NM5">ComplexNormSpace</a>, <font color="Maroon">Y</font> be   <a href="clvect_1.html#NM5">ComplexNormSpace</a>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="clopban1.html#K14" target="_self">C_NormSpace_of_BoundedLinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span>, <font color="Maroon">h</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="clopban1.html#K14" target="_self">C_NormSpace_of_BoundedLinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span>;<br/><b>let </b><font color="Maroon">c</font> be   <a href="cfunct_1.html#NM1">Complex</a>;<br/>





<b>reconsider </b><font color="Maroon">f'</font> = <font color="Maroon">f</font>, <font color="Maroon">h'</font> = <font color="Maroon">h</font> as  <a href="clopban1.html#V3" target="_self">bounded</a> <a href="clopban1.html#NM1">LinearOperator</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">f1</font> = <font color="Maroon">f</font> as   <a href="clvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="clopban1.html#K9" target="_self">C_VectorSpace_of_BoundedLinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon">h1</font> = <font color="Maroon">h</font> as   <a href="clvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="clopban1.html#K9" target="_self">C_VectorSpace_of_BoundedLinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> ;<br/>
<a NAME="E4:70"/>
( <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">f</font> iff <font color="Maroon">h1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">f1</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1"><b>proof </b></a><div class="add">

<div><i><font color="Green">E38</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:70_1_1"/>
<font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">f</font>
 ;<br/><b>hence </b><font color="Maroon">h1</font> = 
<span class="p1">(<span class="default"><a href="csspace.html#K9">Mult_</a> <span class="p2">(<span class="default"><a href="clopban1.html#K8" target="_self">BoundedLinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span>,<span class="p2">(<span class="default"><a href="clopban1.html#K6" target="_self">C_VectorSpace_of_LinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">f</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/>.= 
<font color="Maroon">c</font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">f1</font>
<b>by </b><i><a class="ref" href="clvect_1.html#D1">CLVECT_1:def 1</a></i>
;<br/><br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:70_1_2"/>
<font color="Maroon">h1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">f1</font>
 ;<br/><b>hence </b><font color="Maroon">h</font> = 
<span class="p1">(<span class="default"><a href="csspace.html#K9">Mult_</a> <span class="p2">(<span class="default"><a href="clopban1.html#K8" target="_self">BoundedLinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span>,<span class="p2">(<span class="default"><a href="clopban1.html#K6" target="_self">C_VectorSpace_of_LinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">f1</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/>.= 
<font color="Maroon">c</font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">f</font>
<b>by </b><i><a class="ref" href="clvect_1.html#D1">CLVECT_1:def 1</a></i>
;<br/><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:70_1"/>
( <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">f</font> iff <font color="Maroon">h1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">f1</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:70"/>
( <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">f</font> iff for <font color="Olive">x</font> being  <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font> holds  <font color="Maroon">h</font> <a href="clopban1.html#K15" target="_self">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="clvect_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="clopban1.html#K15" target="_self">.</a> <font color="Olive">x</font></span>)</span> )
 <b>by </b><i/>;<br/>


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