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

<a NAME="E1:14_3_1"/><i><font color="Green">E11</font></i>: 
1 <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> 0
 
;<br/>
<a NAME="E2:14_3_1"/>
1 <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = 1 as  <a href="xcmplx_0.html#V1" target="_self">complex</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  <b>by </b><i><a class="ref" href="xcmplx_0.html#D2" target="_self">Def2</a>, <a class="txt" href="xcmplx_0.html#E15"><i><font color="Green">Lemma10</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E5:14_3_1"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="arytm_0.html#K5">*]</a></span>
 <b>and </b><br/><a NAME="E6:14_3_1"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="arytm_0.html#K5">*]</a></span>
 <b>and </b><br/><a NAME="E7:14_3_1"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="xcmplx_0.html#K3" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2">(<span class="default"><a href="arytm_0.html#K1">+</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K3">opp</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="arytm_0.html#K1">+</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span><a href="arytm_0.html#K5">*]</a></span>
 <b>by </b><i><a class="ref" href="xcmplx_0.html#D5" target="_self">Def5</a></i>;<br/>
<a NAME="E8:14_3_1"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="txt" href="xcmplx_0.html#E5:14_3_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="arytm_0.html#T26">ARYTM_0:26</a></i>;<br/>
<a NAME="E9:14_3_1"/><b>then </b>
 <a href="arytm_0.html#K2">*</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="arytm_0.html#T14">ARYTM_0:14</a></i>;<br/>
<a NAME="E10:14_3_1"/><b>then </b><i><font color="Green">E16</font></i>: 
 <a href="arytm_0.html#K1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="arytm_0.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="ref" href="arytm_0.html#T13">ARYTM_0:13</a></i>;<br/>
<a NAME="E11:14_3_1"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="txt" href="xcmplx_0.html#E5:14_3_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="xcmplx_0.html#E8:14_3_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="arytm_0.html#D7">ARYTM_0:def 7</a></i>;<br/>
<a NAME="E12:14_3_1"/>
 <a href="arytm_0.html#K1">+</a> 0,<span class="p1">(<span class="default"><a href="arytm_0.html#K3">opp</a> 0</span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="arytm_0.html#D4">ARYTM_0:def 4</a></i>;<br/>
<a NAME="E13:14_3_1"/><b>then </b><i><font color="Green">E18</font></i>: 
 <a href="arytm_0.html#K3">opp</a> 0 <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="arytm_0.html#T13">ARYTM_0:13</a></i>;<br/>
<i><font color="Green">E19</font></i>:  <a href="arytm_0.html#K1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K3">opp</a> <span class="p2">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> = 
 <a href="arytm_0.html#K1">+</a> <font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K3">opp</a> <span class="p2">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="xcmplx_0.html#E11:14_3_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="arytm_0.html#T21">ARYTM_0:21</a></i>
<br/>.= 
 <a href="arytm_0.html#K1">+</a> <font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2">*</a> <span class="p2">(<span class="default"><a href="arytm_0.html#K3">opp</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="arytm_0.html#T17">ARYTM_0:17</a></i>
<br/>.= 
 <a href="arytm_0.html#K1">+</a> <font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2">*</a> 0,<font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="xcmplx_0.html#E5:14_3_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="xcmplx_0.html#E13:14_3_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="arytm_0.html#T26">ARYTM_0:26</a></i>
<br/>.= 
 <a href="arytm_0.html#K1">+</a> <font color="Maroon">c<sub>7</sub></font>,0
<b>by </b><i><a class="ref" href="arytm_0.html#T14">ARYTM_0:14</a></i>
<br/>.= 
<font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="ref" href="arytm_0.html#T13">ARYTM_0:13</a></i>
;<br/>
<a NAME="E15:14_3_1"/>
0 <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = 0 as  <a href="xcmplx_0.html#V1" target="_self">complex</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  <b>by </b><i><a class="ref" href="xcmplx_0.html#D2" target="_self">Def2</a>, <a class="txt" href="xcmplx_0.html#E15"><i><font color="Green">Lemma10</font></i></a></i>;<br/>
<b>consider </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> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E18:14_3_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K5" target="_self">"</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="arytm_0.html#K5">*]</a></span>
 <b>and </b><br/><a NAME="E19:14_3_1"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font></span><a href="arytm_0.html#K5">*]</a></span>
 <b>and </b><br/><a NAME="E20:14_3_1"/><i><font color="Green">E21</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K5" target="_self">"</a> </span>)</span> <a href="xcmplx_0.html#K3" target="_self">*</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2">(<span class="default"><a href="arytm_0.html#K1">+</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K3">opp</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="arytm_0.html#K1">+</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>13</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span></span><a href="arytm_0.html#K5">*]</a></span>
 <b>by </b><i><a class="ref" href="xcmplx_0.html#D5" target="_self">Def5</a></i>;<br/>
<a NAME="E21:14_3_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="txt" href="xcmplx_0.html#E19:14_3_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="arytm_0.html#T26">ARYTM_0:26</a></i>;<br/>
<a NAME="E22:14_3_1"/><b>then </b><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="txt" href="xcmplx_0.html#E19:14_3_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="arytm_0.html#D7">ARYTM_0:def 7</a></i>;<br/>
<b>then </b><i><font color="Green">E24</font></i>:  <a href="arytm_0.html#K1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K3">opp</a> <span class="p2">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> = 
 <a href="arytm_0.html#K1">+</a> 0,<span class="p1">(<span class="default"><a href="arytm_0.html#K3">opp</a> <span class="p2">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="arytm_0.html#T14">ARYTM_0:14</a></i>
<br/>.= 
 <a href="arytm_0.html#K3">opp</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>13</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="arytm_0.html#T13">ARYTM_0:13</a></i>
<br/>.= 
0
<b>by </b><i><a class="txt" href="xcmplx_0.html#E13:14_3_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="xcmplx_0.html#E21:14_3_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="arytm_0.html#T14">ARYTM_0:14</a></i>
;<br/>
<i><font color="Green">E25</font></i>:  <a href="arytm_0.html#K1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>13</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span> = 
 <a href="arytm_0.html#K1">+</a> 0,<span class="p1">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="xcmplx_0.html#E21:14_3_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="arytm_0.html#T14">ARYTM_0:14</a></i>
<br/>.= 
 <a href="arytm_0.html#K2">*</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font>
<b>by </b><i><a class="ref" href="arytm_0.html#T13">ARYTM_0:13</a></i>
<br/>.= 
0
<b>by </b><i><a class="txt" href="xcmplx_0.html#E22:14_3_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="arytm_0.html#T14">ARYTM_0:14</a></i>
;<br/>
<b>assume </b><a NAME="E25:14_3_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K3" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xcmplx_0.html#D3" target="_self">XCMPLX_0:def 3</a><br/>

<i><font color="Green">E27</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K5" target="_self">"</a> </span>)</span> <a href="xcmplx_0.html#K3" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K3" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K5" target="_self">"</a> </span>)</span> <a href="xcmplx_0.html#K3" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K3" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="xcmplx_0.html#E14"><i><font color="Green">Lemma9</font></i></a></i>
<br/>.= 
0
<b>by </b><i><a class="txt" href="xcmplx_0.html#E20:14_3_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="xcmplx_0.html#E23:14_3_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="xcmplx_0.html#E24:14_3_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="xcmplx_0.html#E25:14_3_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="arytm_0.html#D7">ARYTM_0:def 7</a></i>
;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K5" target="_self">"</a> </span>)</span> <a href="xcmplx_0.html#K3" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K3" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font> = 
<font color="Maroon">c<sub>4</sub></font> <a href="xcmplx_0.html#K3" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="xcmplx_0.html#D7" target="_self">Def7</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="xcmplx_0.html#E6:14_3_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="xcmplx_0.html#E7:14_3_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="xcmplx_0.html#E10:14_3_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="xcmplx_0.html#E11:14_3_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="xcmplx_0.html#E14:14_3_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="arytm_0.html#T21">ARYTM_0:21</a></i>
;<br/>
<b>hence </b><a NAME="E28:14_3_1"/>
contradiction
 <b>by </b><i><a class="txt" href="xcmplx_0.html#E26:14_3_1"><i><font color="Green">E27</font></i></a></i>;<br/>


</div>
