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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="clvect_1.html#NM2">ComplexLinearSpace</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:68"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>and </b><br/><a NAME="E2:68"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="clvect_1.html#V3" target="_self">lineary-closed</a>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>2</sub></font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="txt" href="clvect_1.html#E1:68"><i><font color="Green">E41</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> =  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="clvect_1.html#E2:68"><i><font color="Green">E42</font></i></a>, <a class="ref" href="clvect_1.html#T21" target="_self">Th21</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> = the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font> <a href="realset1.html#K1">||</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> = the <a href="clvect_1.html#U1" target="_self">Mult</a> of <font color="Maroon">c<sub>1</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K2">COMPLEX</a> ,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<a NAME="E5:68"/>
 <a href="relat_1.html#K1">dom</a> the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</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>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E6:68"/><b>then </b>
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font> <a href="realset1.html#K1">||</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</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>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K3">/\</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</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>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> )
 
<b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a>, <a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/>
<a NAME="E7:68"/><b>then </b><i><font color="Green">E43</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font> <a href="realset1.html#K1">||</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E8:68"/>
 <a href="relat_1.html#K1">dom</a> the <a href="clvect_1.html#U1" target="_self">Mult</a> of <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K2">COMPLEX</a> ,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E9:68"/><b>then </b>
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="clvect_1.html#U1" target="_self">Mult</a> of <font color="Maroon">c<sub>1</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K2">COMPLEX</a> ,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K2">COMPLEX</a> ,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K3">/\</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K2">COMPLEX</a> ,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K2">COMPLEX</a> ,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K2">COMPLEX</a> ,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> )
 
<b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a>, <a class="ref" href="zfmisc_1.html#T118">ZFMISC_1:118</a></i>;<br/>
<a NAME="E10:68"/><b>then </b><i><font color="Green">E44</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="clvect_1.html#U1" target="_self">Mult</a> of <font color="Maroon">c<sub>1</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K2">COMPLEX</a> ,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K2">COMPLEX</a> ,<font color="Maroon">c<sub>3</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E11:68"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font> <a href="realset1.html#K1">||</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>thus </b><a NAME="E1:68_1_1"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> implies ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font> <a href="realset1.html#K1">||</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font> <a href="realset1.html#K1">||</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:68_1_1_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>4</sub></font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E3:68_1_1_1"/><i><font color="Green">E47</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</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>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> )
 
<b>by </b><i><a class="txt" href="clvect_1.html#E1:68_1_1_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font> <a href="realset1.html#K1">||</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span> = 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</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>4</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">c<sub>10</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>9</sub></font>

<br/>.= 
<font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
;<br/>
<b>hence </b><a NAME="E5:68_1_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font> <a href="realset1.html#K1">||</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font> <a href="realset1.html#K1">||</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="clvect_1.html#E7:68"><i><font color="Green">E43</font></i></a>, <a class="txt" href="clvect_1.html#E3:68_1_1_1"><i><font color="Green">E47</font></i></a></i>;<br/>


</div><b>end;</b></div><b>given </b><font color="Maroon">c<sub>9</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E2:68_1_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font> <a href="realset1.html#K1">||</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E3:68_1_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font> <a href="realset1.html#K1">||</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><b>consider </b><font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:68_1_1"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>and </b><br/><a NAME="E6:68_1_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="txt" href="clvect_1.html#E7:68"><i><font color="Green">E43</font></i></a>, <a class="txt" href="clvect_1.html#E2:68_1_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/><a NAME="E7:68_1_1"/><i><font color="Green">E50</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</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>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> )
 <b>by </b><i><a class="txt" href="clvect_1.html#E5:68_1_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>11</sub></font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="clvect_1.html#E5:68_1_1"><i><font color="Green">E48</font></i></a></i>;<br/><font color="Maroon">c<sub>8</sub></font> = 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</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>,<font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="clvect_1.html#E3:68_1_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="clvect_1.html#E6:68_1_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="clvect_1.html#E7:68_1_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">c<sub>12</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>13</sub></font>

;<br/><b>hence </b><a NAME="E10:68_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="clvect_1.html#E2:68"><i><font color="Green">E42</font></i></a>, <a class="txt" href="clvect_1.html#E5:68_1_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="clvect_1.html#D3" target="_self">Def3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:68_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font> <a href="realset1.html#K1">||</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:68"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default">the <a href="clvect_1.html#U1" target="_self">Mult</a> of <font color="Maroon">c<sub>1</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K2">COMPLEX</a> ,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>thus </b><a NAME="E1:68_2_1"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> implies ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="clvect_1.html#U1" target="_self">Mult</a> of <font color="Maroon">c<sub>1</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K2">COMPLEX</a> ,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">the <a href="clvect_1.html#U1" target="_self">Mult</a> of <font color="Maroon">c<sub>1</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K2">COMPLEX</a> ,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_2_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:68_2_1_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E3:68_2_1_1"/><i><font color="Green">E48</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><a href="complex1.html#K6">1r</a> ,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K2">COMPLEX</a> ,<font color="Maroon">c<sub>3</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><a href="complex1.html#K6">1r</a> ,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K2">COMPLEX</a> ,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> )
 
<b>by </b><i><a class="txt" href="clvect_1.html#E1:68_2_1_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default">the <a href="clvect_1.html#U1" target="_self">Mult</a> of <font color="Maroon">c<sub>1</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K2">COMPLEX</a> ,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><a href="complex1.html#K6">1r</a> ,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span> = 
the <a href="clvect_1.html#U1" target="_self">Mult</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><a href="complex1.html#K6">1r</a> ,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<a href="complex1.html#K6">1r</a>  <a href="clvect_1.html#K1" target="_self">*</a> <font color="Maroon">c<sub>9</sub></font>

<br/>.= 
<font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="ref" href="clvect_1.html#D2" target="_self">Def2</a></i>
;<br/>
<b>hence </b><a NAME="E5:68_2_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="clvect_1.html#U1" target="_self">Mult</a> of <font color="Maroon">c<sub>1</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K2">COMPLEX</a> ,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">the <a href="clvect_1.html#U1" target="_self">Mult</a> of <font color="Maroon">c<sub>1</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K2">COMPLEX</a> ,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="clvect_1.html#E10:68"><i><font color="Green">E44</font></i></a>, <a class="txt" href="clvect_1.html#E3:68_2_1_1"><i><font color="Green">E48</font></i></a></i>;<br/>


</div><b>end;</b></div><b>given </b><font color="Maroon">c<sub>9</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E2:68_2_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="clvect_1.html#U1" target="_self">Mult</a> of <font color="Maroon">c<sub>1</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K2">COMPLEX</a> ,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>
 <b>and </b><br/><a NAME="E3:68_2_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">the <a href="clvect_1.html#U1" target="_self">Mult</a> of <font color="Maroon">c<sub>1</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K2">COMPLEX</a> ,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><b>consider </b><font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:68_2_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K2">COMPLEX</a> 
 <b>and </b><br/><a NAME="E6:68_2_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E7:68_2_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="txt" href="clvect_1.html#E10:68"><i><font color="Green">E44</font></i></a>, <a class="txt" href="clvect_1.html#E2:68_2_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/><a NAME="E8:68_2_1"/><i><font color="Green">E52</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K2">COMPLEX</a> ,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K2">COMPLEX</a> ,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> )
 <b>by </b><i><a class="txt" href="clvect_1.html#E5:68_2_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="clvect_1.html#E6:68_2_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>11</sub></font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="clvect_1.html#E6:68_2_1"><i><font color="Green">E50</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>10</sub></font> as   <a href="cfunct_1.html#NM1">Complex</a> <b>by </b><i><a class="txt" href="clvect_1.html#E5:68_2_1"><i><font color="Green">E49</font></i></a></i>;<br/><font color="Maroon">c<sub>8</sub></font> = 
the <a href="clvect_1.html#U1" target="_self">Mult</a> of <font color="Maroon">c<sub>1</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>,<font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="clvect_1.html#E3:68_2_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="clvect_1.html#E7:68_2_1"><i><font color="Green">E51</font></i></a>, <a class="txt" href="clvect_1.html#E8:68_2_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">c<sub>13</sub></font> <a href="clvect_1.html#K1" target="_self">*</a> <font color="Maroon">c<sub>12</sub></font>

;<br/><b>hence </b><a NAME="E12:68_2_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="clvect_1.html#E2:68"><i><font color="Green">E42</font></i></a>, <a class="txt" href="clvect_1.html#E6:68_2_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="clvect_1.html#D3" target="_self">Def3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:68_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default">the <a href="clvect_1.html#U1" target="_self">Mult</a> of <font color="Maroon">c<sub>1</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K2">COMPLEX</a> ,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font> <a href="realset1.html#K1">||</a> <font color="Maroon">c<sub>2</sub></font> as   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>,<font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="clvect_1.html#E7:68"><i><font color="Green">E43</font></i></a>, <a class="txt" href="clvect_1.html#E11:68"><i><font color="Green">E45</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = the <a href="clvect_1.html#U1" target="_self">Mult</a> of <font color="Maroon">c<sub>1</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K2">COMPLEX</a> ,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> as   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K2">COMPLEX</a> ,<font color="Maroon">c<sub>3</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>,<font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="clvect_1.html#E10:68"><i><font color="Green">E44</font></i></a>, <a class="txt" href="clvect_1.html#E12:68"><i><font color="Green">E46</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>10</sub></font> =  <a href="clvect_1.html#G1" target="_self">CLSStruct</a>(# <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> #);<br/>
<a NAME="E15:68"/>
(  <a href="clvect_1.html#G1" target="_self">CLSStruct</a>(# <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> #) is    <a href="clvect_1.html#M1" target="_self">Subspace</a> of <font color="Maroon">c<sub>1</sub></font> &amp; the <a href="struct_0.html#U1">carrier</a> of <a href="clvect_1.html#G1" target="_self">CLSStruct</a>(# <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> #) <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="ref" href="clvect_1.html#T44" target="_self">Th44</a></i>;<br/>
<b>hence </b><a NAME="E16:68"/>
ex <font color="Olive">b<sub>1</sub></font> being <a href="clvect_1.html#V1" target="_self">strict</a>  <a href="clvect_1.html#M1" target="_self">Subspace</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">b<sub>1</sub></font>
 ;<br/>


</div>
