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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="rlvect_1.html#NM2">RealLinearSpace</a>;<br/><b>let </b><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> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font>, <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="real_1.html#NM1">Real</a>;<br/>

















<b>assume </b><a NAME="E1:18"/><i><font color="Green">E16</font></i>: 
( not  <a href="anproj_1.html#R1" target="_self">are_Prop</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>2</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="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  )
 ;<br/>

<a NAME="E2:18"/><b>then </b><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>7</sub></font>
 
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/><a NAME="E2:18_1"/><i><font color="Green">E18</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="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><br/>for <font color="Olive">b<sub>5</sub></font>, <font color="Olive">b<sub>6</sub></font>, <font color="Olive">b<sub>7</sub></font>, <font color="Olive">b<sub>8</sub></font> being  <a href="real_1.html#NM1">Real</a> st not  <a href="anproj_1.html#R1" target="_self">are_Prop</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> &amp; <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>5</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>7</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">b<sub>2</sub></font></span>)</span> &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>6</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>8</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">b<sub>2</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Olive">b<sub>5</sub></font> <a href="real_1.html#K4">*</a> <font color="Olive">b<sub>8</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>6</sub></font> <a href="real_1.html#K4">*</a> <font color="Olive">b<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Olive">b<sub>1</sub></font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Olive">b<sub>2</sub></font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font> holds <br/> <a href="anproj_1.html#R1" target="_self">are_Prop</a> <font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>13</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>14</sub></font>, <font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>16</sub></font>, <font color="Maroon">c<sub>17</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/>















<b>assume </b><b>that </b><br/><a NAME="E1:18_1_1"/><i><font color="Green">E19</font></i>: 
( not  <a href="anproj_1.html#R1" target="_self">are_Prop</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="rlvect_1.html#K3">*</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>16</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="rlvect_1.html#K3">*</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>17</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>10</sub></font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">c<sub>11</sub></font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  )
 <b>and </b><br/><a NAME="E2:18_1_1"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>and </b><br/><a NAME="E3:18_1_1"/><i><font color="Green">E21</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/>

0 = 
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>16</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="anproj_1.html#E1:18_1_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="anproj_1.html#E2:18_1_1"><i><font color="Green">E20</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>16</sub></font>

;<br/>
<a NAME="E5:18_1_1"/><b>then </b><i><font color="Green">E22</font></i>: 
(  <a href="real_1.html#K1">-</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i><a class="ref" href="xcmplx_1.html#T6">XCMPLX_1:6</a></i>;<br/>
<div><i><font color="Green">E23</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_1_1_2"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><b>then </b><font color="Maroon">c<sub>12</sub></font> = 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default">0 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="anproj_1.html#E1:18_1_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="anproj_1.html#E2:18_1_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="rlvect_1.html#T23">RLVECT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <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>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T23">RLVECT_1:23</a></i>
;<br/><b>hence </b><a NAME="E3:18_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="anproj_1.html#E3:18_1_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>;<br/></div><b>end;</b></div>
<a NAME="E7:18_1_1"/><b>then </b><i><font color="Green">E24</font></i>: 
( <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#NR2" target="_self">&lt;&gt;</a> 0 )
 
<b>by </b><i><a class="txt" href="anproj_1.html#E5:18_1_1"><i><font color="Green">E22</font></i></a></i>;<br/>
<a NAME="E8:18_1_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="real_1.html#K2">"</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="anproj_1.html#E6:18_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="xcmplx_1.html#T203">XCMPLX_1:203</a></i>;<br/>
<a NAME="E9:18_1_1"/>
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<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> )
 
<b>by </b><i><a class="txt" href="anproj_1.html#E1:18_1_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="anproj_1.html#E2:18_1_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="anproj_1.html#E7:18_1_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="rlvect_1.html#T23">RLVECT_1:23</a></i>;<br/>
<a NAME="E10:18_1_1"/><b>then </b>
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font> )
 
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>;<br/>
<a NAME="E11:18_1_1"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="anproj_1.html#E6:18_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="analoaf.html#T12">ANALOAF:12</a></i>;<br/>
<a NAME="E12:18_1_1"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="real_1.html#K2">"</a> </span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>;<br/>
<a NAME="E13:18_1_1"/><b>then </b><i><font color="Green">E26</font></i>: 
( 1 <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"><font color="Maroon">c<sub>17</sub></font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="real_1.html#K2">"</a> </span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>12</sub></font> &amp; 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_1_1_3"/>
<font color="Maroon">c<sub>17</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:18_1_1_3"/><b>then </b>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="anproj_1.html#E8:18_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="xcmplx_1.html#T6">XCMPLX_1:6</a></i>;<br/><b>then </b><font color="Maroon">c<sub>13</sub></font> = 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default">0 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="anproj_1.html#E1:18_1_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="anproj_1.html#E7:18_1_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="rlvect_1.html#T23">RLVECT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <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>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T23">RLVECT_1:23</a></i>
;<br/><b>hence </b><a NAME="E4:18_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="anproj_1.html#E3:18_1_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E15:18_1_1"/>
 <a href="anproj_1.html#R1" target="_self">are_Prop</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="anproj_1.html#E13:18_1_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="anproj_1.html#D2" target="_self">Def2</a></i>;<br/>


</div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_1_2"/><i><font color="Green">E19</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 ;<br/><div><i><font color="Green">E20</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_1_2_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:18_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="anproj_1.html#E1:18"><i><font color="Green">E16</font></i></a>, <a class="txt" href="anproj_1.html#E1:18_1_2"><i><font color="Green">E19</font></i></a>, <a class="ref" href="xcmplx_1.html#T6">XCMPLX_1:6</a></i>;<br/><a NAME="E3:18_1_2_1"/><b>then </b>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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>1</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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>1</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="anproj_1.html#E1:18"><i><font color="Green">E16</font></i></a>, <a class="txt" href="anproj_1.html#E1:18_1_2_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="rlvect_1.html#T23">RLVECT_1:23</a></i>;<br/><a NAME="E4:18_1_2_1"/><b>then </b><i><font color="Green">E22</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="real_1.html#K2">"</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 <b>by </b><i><a class="txt" href="anproj_1.html#E1:18_1_2"><i><font color="Green">E19</font></i></a>, <a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a>, <a class="ref" href="xcmplx_1.html#T203">XCMPLX_1:203</a></i>;<br/><b>then </b><font color="Maroon">c<sub>5</sub></font> = 
<font color="Maroon">c<sub>8</sub></font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="anproj_1.html#E1:18_1_2"><i><font color="Green">E19</font></i></a>, <a class="ref" href="analoaf.html#T12">ANALOAF:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="real_1.html#K2">"</a> </span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
;<br/><a NAME="E6:18_1_2_1"/><b>then </b>
( 1 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="real_1.html#K2">"</a> </span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 <b>by </b><i><a class="txt" href="anproj_1.html#E1:18_1_2"><i><font color="Green">E19</font></i></a>, <a class="txt" href="anproj_1.html#E4:18_1_2_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a>, <a class="ref" href="xcmplx_1.html#T6">XCMPLX_1:6</a></i>;<br/><b>hence </b><a NAME="E7:18_1_2_1"/>
 <a href="anproj_1.html#R1" target="_self">are_Prop</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="anproj_1.html#D2" target="_self">Def2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_1_2_2"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><a NAME="E2:18_1_2_2"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="anproj_1.html#E1:18"><i><font color="Green">E16</font></i></a>, <a class="txt" href="anproj_1.html#E19"><i><font color="Green">Lemma13</font></i></a></i>;<br/><a NAME="E3:18_1_2_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="txt" href="anproj_1.html#E2:18"><i><font color="Green">E17</font></i></a>, <a class="txt" href="anproj_1.html#E1:18_1_2"><i><font color="Green">E19</font></i></a>, <a class="txt" href="anproj_1.html#E1:18_1_2_2"><i><font color="Green">E21</font></i></a>, <a class="ref" href="xcmplx_1.html#T6">XCMPLX_1:6</a></i>;<br/><b>hence </b><a NAME="E4:18_1_2_2"/>
 <a href="anproj_1.html#R1" target="_self">are_Prop</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="anproj_1.html#E2:18"><i><font color="Green">E17</font></i></a>, <a class="txt" href="anproj_1.html#E1:18_1_2_2"><i><font color="Green">E21</font></i></a>, <a class="txt" href="anproj_1.html#E2:18_1_2_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="anproj_1.html#D2" target="_self">Def2</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:18_1_2"/>
(  <a href="anproj_1.html#R1" target="_self">are_Prop</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font> or <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="anproj_1.html#E2:18_1_2"><i><font color="Green">E20</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:18_1"/>
(  <a href="anproj_1.html#R1" target="_self">are_Prop</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font> or <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="anproj_1.html#E1:18"><i><font color="Green">E16</font></i></a>, <a class="txt" href="anproj_1.html#E2:18_1"><i><font color="Green">E18</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:18"/>
(  <a href="anproj_1.html#R1" target="_self">are_Prop</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font> or <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/>


</div>
