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

<b>consider </b><font color="Maroon">c<sub>1</sub></font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E2:59"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K9">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K9">}</a></span> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="anproj_2.html#E58"><i><font color="Green">Lemma54</font></i></a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="funcsdom.html#K11">RealVectSpace</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>consider </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>, <font color="Maroon">c<sub>10</sub></font> being    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E4:59"/><i><font color="Green">E57</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>, <font color="Olive">b<sub>4</sub></font> being  <a href="real_1.html#NM1">Real</a> st <span class="p1">(<span class="default"><a href="funcsdom.html#K6">RealFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funcsdom.html#K6">RealFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funcsdom.html#K6">RealFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="funcsdom.html#K8">RealFuncExtMult</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p4"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="funcsdom.html#K8">RealFuncExtMult</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p4"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funcsdom.html#K8">RealFuncExtMult</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>3</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funcsdom.html#K8">RealFuncExtMult</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>4</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="funct_2.html#R2">=</a>  <a href="funcsdom.html#K9">RealFuncZero</a> <font color="Maroon">c<sub>1</sub></font> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0 ) ) &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K1">REAL</a> ex <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font>, <font color="Olive">b<sub>5</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="funcsdom.html#K6">RealFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funcsdom.html#K6">RealFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funcsdom.html#K6">RealFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="funcsdom.html#K8">RealFuncExtMult</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p4"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="funcsdom.html#K8">RealFuncExtMult</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p4"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>3</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funcsdom.html#K8">RealFuncExtMult</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>4</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funcsdom.html#K8">RealFuncExtMult</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>5</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span> ) )
 <b>by </b><i><a class="txt" href="anproj_2.html#E2:59"><i><font color="Green">E56</font></i></a>, <a class="ref" href="anproj_2.html#T22" target="_self">Th22</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>10</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="funcsdom.html#K11">RealVectSpace</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> ;<br/>
<a NAME="E6:59"/><i><font color="Green">E58</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>, <font color="Olive">b<sub>4</sub></font> being  <a href="real_1.html#NM1">Real</a> st <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>4</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="funcsdom.html#K11">RealVectSpace</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0 )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>16</sub></font>, <font color="Maroon">c<sub>17</sub></font>, <font color="Maroon">c<sub>18</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E1:59_1"/><i><font color="Green">E59</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="funcsdom.html#K11">RealVectSpace</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>15</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>16</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>17</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>13</sub></font>, <font color="Maroon">c<sub>22</sub></font> = <font color="Maroon">c<sub>18</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>14</sub></font> as    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K1">REAL</a>  ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>23</sub></font> = <font color="Maroon">c<sub>19</sub></font>, <font color="Maroon">c<sub>24</sub></font> = <font color="Maroon">c<sub>20</sub></font>, <font color="Maroon">c<sub>25</sub></font> = <font color="Maroon">c<sub>21</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="funcsdom.html#K11">RealVectSpace</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> ;<br/>
 <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="funcsdom.html#K11">RealVectSpace</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> = 
the <a href="rlvect_1.html#U1">add</a> of <span class="p1">(<span class="default"><a href="funcsdom.html#K11">RealVectSpace</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <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"><span class="p3">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>25</sub></font></span>)</span>,<font color="Maroon">c<sub>22</sub></font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="txt" href="anproj_2.html#E1:59_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="rlvect_1.html#D3">RLVECT_1:def 3</a></i>
<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <span class="p1">(<span class="default"><a href="funcsdom.html#K11">RealVectSpace</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <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="rlvect_1.html#U1">add</a> of <span class="p3">(<span class="default"><a href="funcsdom.html#K11">RealVectSpace</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>24</sub></font></span>)</span>,<font color="Maroon">c<sub>21</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<font color="Maroon">c<sub>22</sub></font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D3">RLVECT_1:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="funcsdom.html#K6">RealFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funcsdom.html#K6">RealFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="funcsdom.html#K6">RealFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p5"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<font color="Maroon">c<sub>21</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<font color="Maroon">c<sub>22</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/>.= 
<span class="p1">(<span class="default"><a href="funcsdom.html#K6">RealFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funcsdom.html#K6">RealFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="funcsdom.html#K6">RealFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font></span>)</span>,<font color="Maroon">c<sub>21</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<font color="Maroon">c<sub>22</sub></font></span><a href="domain_1.html#K1">]</a></span>

<br/>.= 
<span class="p1">(<span class="default"><a href="funcsdom.html#K6">RealFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funcsdom.html#K6">RealFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="funcsdom.html#K6">RealFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font></span>)</span>,<font color="Maroon">c<sub>21</sub></font></span>)</span>,<font color="Maroon">c<sub>22</sub></font></span><a href="domain_1.html#K1">]</a></span>

<br/>.= 
<span class="p1">(<span class="default"><a href="funcsdom.html#K6">RealFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funcsdom.html#K6">RealFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funcsdom.html#K6">RealFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font></span>)</span>,<font color="Maroon">c<sub>21</sub></font></span>)</span>,<font color="Maroon">c<sub>22</sub></font>

<br/>.= 
<span class="p1">(<span class="default"><a href="funcsdom.html#K6">RealFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funcsdom.html#K6">RealFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funcsdom.html#K6">RealFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="funcsdom.html#K8">RealFuncExtMult</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p4"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<font color="Maroon">c<sub>20</sub></font></span>)</span>,<font color="Maroon">c<sub>21</sub></font></span>)</span>,<font color="Maroon">c<sub>22</sub></font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D4">RLVECT_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="funcsdom.html#K6">RealFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funcsdom.html#K6">RealFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funcsdom.html#K6">RealFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="funcsdom.html#K8">RealFuncExtMult</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p4"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="funcsdom.html#K8">RealFuncExtMult</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p4"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span>)</span>,<font color="Maroon">c<sub>21</sub></font></span>)</span>,<font color="Maroon">c<sub>22</sub></font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D4">RLVECT_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="funcsdom.html#K6">RealFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funcsdom.html#K6">RealFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funcsdom.html#K6">RealFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="funcsdom.html#K8">RealFuncExtMult</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p4"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="funcsdom.html#K8">RealFuncExtMult</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p4"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funcsdom.html#K8">RealFuncExtMult</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span>)</span>,<font color="Maroon">c<sub>22</sub></font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D4">RLVECT_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="funcsdom.html#K6">RealFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funcsdom.html#K6">RealFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funcsdom.html#K6">RealFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="funcsdom.html#K8">RealFuncExtMult</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p4"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="funcsdom.html#K8">RealFuncExtMult</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p4"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funcsdom.html#K8">RealFuncExtMult</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funcsdom.html#K8">RealFuncExtMult</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D4">RLVECT_1:def 4</a></i>
;<br/>
<a NAME="E5:59_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="funcsdom.html#K6">RealFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funcsdom.html#K6">RealFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funcsdom.html#K6">RealFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="funcsdom.html#K8">RealFuncExtMult</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p4"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="funcsdom.html#K8">RealFuncExtMult</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p4"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funcsdom.html#K8">RealFuncExtMult</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funcsdom.html#K8">RealFuncExtMult</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="funct_2.html#R2">=</a>  <a href="funcsdom.html#K9">RealFuncZero</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D2">RLVECT_1:def 2</a></i>;<br/>
<b>hence </b><a NAME="E6:59_1"/>
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="txt" href="anproj_2.html#E4:59"><i><font color="Green">E57</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:59"/><i><font color="Green">E59</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="funcsdom.html#K11">RealVectSpace</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>ex <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font>, <font color="Olive">b<sub>5</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>4</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>5</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>15</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="funcsdom.html#K11">RealVectSpace</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>15</sub></font> as    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K1">REAL</a>  ;<br/>
<b>consider </b><font color="Maroon">c<sub>17</sub></font>, <font color="Maroon">c<sub>18</sub></font>, <font color="Maroon">c<sub>19</sub></font>, <font color="Maroon">c<sub>20</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:59_2"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="funcsdom.html#K6">RealFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funcsdom.html#K6">RealFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funcsdom.html#K6">RealFuncAdd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="funcsdom.html#K8">RealFuncExtMult</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p4"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="funcsdom.html#K8">RealFuncExtMult</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p4"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funcsdom.html#K8">RealFuncExtMult</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funcsdom.html#K8">RealFuncExtMult</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="anproj_2.html#E4:59"><i><font color="Green">E57</font></i></a></i>;<br/>
<a NAME="E4:59_2"/>
( <font color="Maroon">c<sub>17</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funcsdom.html#K8">RealFuncExtMult</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funcsdom.html#K8">RealFuncExtMult</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funcsdom.html#K8">RealFuncExtMult</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funcsdom.html#K8">RealFuncExtMult</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="domain_1.html#K1">]</a></span> )
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D4">RLVECT_1:def 4</a></i>;<br/>
<b>then </b><font color="Maroon">c<sub>16</sub></font> = 
the <a href="rlvect_1.html#U1">add</a> of <span class="p1">(<span class="default"><a href="funcsdom.html#K11">RealVectSpace</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <span class="p2">(<span class="default"><a href="funcsdom.html#K11">RealVectSpace</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <span class="p2">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <span class="p3">(<span class="default"><a href="funcsdom.html#K11">RealVectSpace</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="anproj_2.html#E3:59_2"><i><font color="Green">E60</font></i></a></i>
<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <span class="p1">(<span class="default"><a href="funcsdom.html#K11">RealVectSpace</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <span class="p2">(<span class="default"><a href="funcsdom.html#K11">RealVectSpace</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <span class="p4">(<span class="default"><a href="funcsdom.html#K11">RealVectSpace</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p4"><a href="domain_1.html#K1">[</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>

<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <span class="p1">(<span class="default"><a href="funcsdom.html#K11">RealVectSpace</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <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 <span class="p3">(<span class="default"><a href="funcsdom.html#K11">RealVectSpace</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><span class="p4">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <span class="p5">(<span class="default"><a href="funcsdom.html#K11">RealVectSpace</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p5"><a href="domain_1.html#K1">[</a><span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>,<span class="p0">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span>

<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <span class="p1">(<span class="default"><a href="funcsdom.html#K11">RealVectSpace</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <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 <span class="p3">(<span class="default"><a href="funcsdom.html#K11">RealVectSpace</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></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/>.= 
the <a href="rlvect_1.html#U1">add</a> of <span class="p1">(<span class="default"><a href="funcsdom.html#K11">RealVectSpace</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></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/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D3">RLVECT_1:def 3</a></i>
;<br/>
<b>hence </b><a NAME="E6:59_2"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>4</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E8:59"/>
<font color="Maroon">c<sub>11</sub></font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a> 
 
<b>by </b><i><a class="txt" href="anproj_2.html#E6:59"><i><font color="Green">E58</font></i></a>, <a class="ref" href="anproj_2.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E9:59"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="funcsdom.html#K11">RealVectSpace</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D13">RLVECT_1:def 13</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>15</sub></font> =  <a href="funcsdom.html#K11">RealVectSpace</a> <font color="Maroon">c<sub>1</sub></font> as  non <a href="realset2.html#V3">trivial</a> <a href="rlvect_1.html#NM2">RealLinearSpace</a> <b>by </b><i><a class="ref" href="anproj_1.html#D8">ANPROJ_1:def 8</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>15</sub></font>
;<br/>

<b>reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>13</sub></font>, <font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>14</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>15</sub></font> ;<br/>
<b>take </b>
<font color="Maroon">c<sub>16</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>17</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>18</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>19</sub></font>
;<br/>

<b>thus </b><a NAME="E12:59"/>
( ( 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>, <font color="Olive">b<sub>4</sub></font> being  <a href="real_1.html#NM1">Real</a> st <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>17</sub></font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>4</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>15</sub></font> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0 ) ) &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>15</sub></font>ex <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font>, <font color="Olive">b<sub>5</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>17</sub></font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>4</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>5</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> ) )
 <b>by </b><i><a class="txt" href="anproj_2.html#E6:59"><i><font color="Green">E58</font></i></a>, <a class="txt" href="anproj_2.html#E7:59"><i><font color="Green">E59</font></i></a></i>;<br/>


</div>
