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

<b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="csspace.html#L1" target="_self">CUNITSTR</a> ;<br/>

<b>assume </b><a NAME="E1:115"/><i><font color="Green">E73</font></i>: 
 <a href="clvect_1.html#G1">CLSStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">c<sub>4</sub></font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>4</sub></font>,the <a href="clvect_1.html#U1">Mult</a> of <font color="Maroon">c<sub>4</sub></font> #) is   <a href="clvect_1.html#NM2">ComplexLinearSpace</a>
 ;<br/>

<a NAME="E2:115"/>
not the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i><a class="txt" href="csspace.html#E1:115"><i><font color="Green">E73</font></i></a>, <a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as  non <a href="struct_0.html#V3">empty</a>  <a href="clvect_1.html#L1">CLSStruct</a>  <b>by </b><i><a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> =  <a href="clvect_1.html#G1">CLSStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">c<sub>5</sub></font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>5</sub></font>,the <a href="clvect_1.html#U1">Mult</a> of <font color="Maroon">c<sub>5</sub></font> #) as   <a href="clvect_1.html#NM2">ComplexLinearSpace</a> <b>by </b><i><a class="txt" href="csspace.html#E1:115"><i><font color="Green">E73</font></i></a></i>;<br/>
<a NAME="E5:115"/><i><font color="Green">E74</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>5</sub></font> holds <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> be   <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>5</sub></font>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>6</sub></font> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>6</sub></font> ;<br/>
<b>thus </b><font color="Maroon">c<sub>7</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>8</sub></font> = 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>5</sub></font> <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>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="domain_1.html#K1">]</a></span>

<br/>.= 
<font color="Maroon">c<sub>9</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>10</sub></font>

<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>5</sub></font> <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>10</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D3">RLVECT_1:def 3</a></i>
<br/>.= 
<font color="Maroon">c<sub>8</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>7</sub></font>

;<br/>


</div><b>end;</b></div>
<a NAME="E6:115"/><i><font color="Green">E75</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>5</sub></font> holds <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b<sub>3</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> be   <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>5</sub></font>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>6</sub></font> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>6</sub></font> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>6</sub></font> ;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font> = 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<font color="Maroon">c<sub>9</sub></font></span><a href="domain_1.html#K1">]</a></span>

<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <span class="p3"><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></span>)</span>,<font color="Maroon">c<sub>9</sub></font></span><a href="domain_1.html#K1">]</a></span>

<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>,<font color="Maroon">c<sub>9</sub></font></span><a href="domain_1.html#K1">]</a></span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>12</sub></font>

<br/>.= 
<font color="Maroon">c<sub>10</sub></font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>5</sub></font> <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>10</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span>

<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<span class="p2">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span><a href="tarski.html#K4">]</a></span>

<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span>

<br/>.= 
<font color="Maroon">c<sub>7</sub></font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>

;<br/>


</div><b>end;</b></div>
<a NAME="E7:115"/><i><font color="Green">E76</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>5</sub></font> holds <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>5</sub></font>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>6</sub></font> ;<br/>
<b>thus </b><font color="Maroon">c<sub>7</sub></font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> = 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span>

<br/>.= 
<font color="Maroon">c<sub>8</sub></font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>

<br/>.= 
<font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D7">RLVECT_1:def 7</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E8:115"/><i><font color="Green">E77</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>5</sub></font>ex <font color="Olive">b<sub>2</sub></font> being  <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>5</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>5</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>5</sub></font>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>6</sub></font> ;<br/>
<b>consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>6</sub></font><b> such that </b><br/><a NAME="E3:115_4"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D8">RLVECT_1:def 8</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>5</sub></font> ;<br/>
<i><font color="Green">E79</font></i>: <font color="Maroon">c<sub>7</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>10</sub></font> = 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</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>10</sub></font></span><a href="domain_1.html#K1">]</a></span>

<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="csspace.html#E3:115_4"><i><font color="Green">E78</font></i></a></i>
;<br/>
<b>take </b>
<font color="Maroon">c<sub>10</sub></font>
;<br/>

<b>thus </b><a NAME="E6:115_4"/>
<font color="Maroon">c<sub>7</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="csspace.html#E5:115_4"><i><font color="Green">E79</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:115"/><i><font color="Green">E78</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="cfunct_1.html#NM1">Complex</a><br/>for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>5</sub></font> holds <font color="Olive">b<sub>1</sub></font> <a href="clvect_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Olive">b<sub>3</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="cfunct_1.html#NM1">Complex</a>;<br/><b>let </b><font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> be   <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>5</sub></font>;<br/>



<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>6</sub></font> ;<br/>
<b>thus </b><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>8</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> = 
the <a href="clvect_1.html#U1">Mult</a> of <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span>

<br/>.= 
the <a href="clvect_1.html#U1">Mult</a> of <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<span class="p2">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span><a href="tarski.html#K4">]</a></span>

<br/>.= 
the <a href="clvect_1.html#U1">Mult</a> of <font color="Maroon">c<sub>5</sub></font> <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>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span>

<br/>.= 
<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>10</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="clvect_1.html#D2">CLVECT_1:def 2</a></i>
<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span>

<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default">the <a href="clvect_1.html#U1">Mult</a> of <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>

<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default">the <a href="clvect_1.html#U1">Mult</a> of <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<span class="p2">(<span class="default">the <a href="clvect_1.html#U1">Mult</a> of <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span><a href="tarski.html#K4">]</a></span>

<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p2">(<span class="default">the <a href="clvect_1.html#U1">Mult</a> of <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span><a href="domain_1.html#K1">]</a></span>

<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>

;<br/>


</div><b>end;</b></div>
<a NAME="E10:115"/><i><font color="Green">E79</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="cfunct_1.html#NM1">Complex</a><br/>for <font color="Olive">b<sub>3</sub></font> being  <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>5</sub></font> holds <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="complex1.html#K8">+</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="clvect_1.html#K1">*</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Olive">b<sub>3</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> be   <a href="cfunct_1.html#NM1">Complex</a>;<br/><b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>5</sub></font>;<br/>



<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>6</sub></font> ;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="complex1.html#K8">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="clvect_1.html#K1">*</a> <font color="Maroon">c<sub>9</sub></font> = 
the <a href="clvect_1.html#U1">Mult</a> of <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="complex1.html#K8">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<font color="Maroon">c<sub>9</sub></font></span><a href="domain_1.html#K1">]</a></span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="complex1.html#K8">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="clvect_1.html#K1">*</a> <font color="Maroon">c<sub>10</sub></font>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="clvect_1.html#D2">CLVECT_1:def 2</a></i>
<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span>

<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default">the <a href="clvect_1.html#U1">Mult</a> of <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>

<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default">the <a href="clvect_1.html#U1">Mult</a> of <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<span class="p2">(<span class="default">the <a href="clvect_1.html#U1">Mult</a> of <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span><a href="tarski.html#K4">]</a></span>

<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>,<span class="p2">(<span class="default">the <a href="clvect_1.html#U1">Mult</a> of <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span><a href="domain_1.html#K1">]</a></span>

<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>

;<br/>


</div><b>end;</b></div>
<a NAME="E11:115"/><i><font color="Green">E80</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="cfunct_1.html#NM1">Complex</a><br/>for <font color="Olive">b<sub>3</sub></font> being  <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>5</sub></font> holds <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="complex1.html#K9">*</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="clvect_1.html#K1">*</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="clvect_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Olive">b<sub>3</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> be   <a href="cfunct_1.html#NM1">Complex</a>;<br/><b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>5</sub></font>;<br/>



<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>6</sub></font> ;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="complex1.html#K9">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="clvect_1.html#K1">*</a> <font color="Maroon">c<sub>9</sub></font> = 
the <a href="clvect_1.html#U1">Mult</a> of <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="complex1.html#K9">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<font color="Maroon">c<sub>9</sub></font></span><a href="domain_1.html#K1">]</a></span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="complex1.html#K9">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="clvect_1.html#K1">*</a> <font color="Maroon">c<sub>10</sub></font>

<br/>.= 
<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>8</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="clvect_1.html#D2">CLVECT_1:def 2</a></i>
<br/>.= 
the <a href="clvect_1.html#U1">Mult</a> of <font color="Maroon">c<sub>5</sub></font> <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>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span>

<br/>.= 
the <a href="clvect_1.html#U1">Mult</a> of <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<span class="p2">(<span class="default">the <a href="clvect_1.html#U1">Mult</a> of <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span><a href="tarski.html#K4">]</a></span>

<br/>.= 
the <a href="clvect_1.html#U1">Mult</a> of <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span>

<br/>.= 
<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>8</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>

;<br/>


</div><b>end;</b></div>
<a NAME="E12:115"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>5</sub></font> holds <a href="complex1.html#K6">1r</a>  <a href="clvect_1.html#K1">*</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>5</sub></font>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>6</sub></font> ;<br/>
<b>thus </b><a href="complex1.html#K6">1r</a>  <a href="clvect_1.html#K1">*</a> <font color="Maroon">c<sub>7</sub></font> = 
the <a href="clvect_1.html#U1">Mult</a> of <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><a href="complex1.html#K6">1r</a> ,<font color="Maroon">c<sub>7</sub></font></span><a href="domain_1.html#K1">]</a></span>

<br/>.= 
<a href="complex1.html#K6">1r</a>  <a href="clvect_1.html#K1">*</a> <font color="Maroon">c<sub>8</sub></font>

<br/>.= 
<font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="ref" href="clvect_1.html#D2">CLVECT_1:def 2</a></i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E13:115"/>
<font color="Maroon">c<sub>4</sub></font> is   <a href="clvect_1.html#NM2">ComplexLinearSpace</a>
 <b>by </b><i><a class="txt" href="csspace.html#E5:115"><i><font color="Green">E74</font></i></a>, <a class="txt" href="csspace.html#E6:115"><i><font color="Green">E75</font></i></a>, <a class="txt" href="csspace.html#E7:115"><i><font color="Green">E76</font></i></a>, <a class="txt" href="csspace.html#E8:115"><i><font color="Green">E77</font></i></a>, <a class="txt" href="csspace.html#E9:115"><i><font color="Green">E78</font></i></a>, <a class="txt" href="csspace.html#E10:115"><i><font color="Green">E79</font></i></a>, <a class="txt" href="csspace.html#E11:115"><i><font color="Green">E80</font></i></a>, <a class="ref" href="clvect_1.html#T1">CLVECT_1:1</a></i>;<br/>


</div>
