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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="clvect_1.html#NM5">ComplexNormSpace</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="csspace4.html#K9" target="_self">C_NormSpace_of_BoundedFunctions</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be  <a href="csspace4.html#V1" target="_self">bounded</a> <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:48"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E2:48"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="cfunct_1.html#NM1">Complex</a>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="clvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="csspace4.html#K5" target="_self">C_VectorSpace_of_BoundedFunctions</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="clvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="csspace4.html#K5" target="_self">C_VectorSpace_of_BoundedFunctions</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> ;<br/>
<a NAME="E5:48"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font> iff <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">c<sub>8</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1"><b>proof </b></a><div class="add">

<div><i><font color="Green">E39</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:48_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><b>hence </b><font color="Maroon">c<sub>9</sub></font> = 
<span class="p1">(<span class="default"><a href="csspace.html#K9">Mult_</a> <span class="p2">(<span class="default"><a href="csspace4.html#K4" target="_self">ComplexBoundedFunctions</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="clopban1.html#K3">ComplexVectSpace</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>7</sub></font>,<font color="Maroon">c<sub>3</sub></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<sub>7</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">c<sub>8</sub></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="48_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:48_1_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><b>hence </b><font color="Maroon">c<sub>4</sub></font> = 
<span class="p1">(<span class="default"><a href="csspace.html#K9">Mult_</a> <span class="p2">(<span class="default"><a href="csspace4.html#K4" target="_self">ComplexBoundedFunctions</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="clopban1.html#K3">ComplexVectSpace</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></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<sub>7</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></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="E3:48_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font> iff <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="csspace4.html#E1:48_1"><i><font color="Green">E39</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:48"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font> iff for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="clvect_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="csspace4.html#E1:48"><i><font color="Green">E37</font></i></a>, <a class="txt" href="csspace4.html#E2:48"><i><font color="Green">E38</font></i></a>, <a class="ref" href="csspace4.html#T12" target="_self">Th12</a></i>;<br/>


</div>
