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<a NAME="E1:14_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">j</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>, </i>;<br/>
<b>consider </b><font color="Maroon">u1</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">u2</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">v1</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">v2</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="E4:14_3_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">u1</font>,<font color="Maroon">u2</font></span><a href="arytm_0.html#K5">*]</a></span>
 <b>and </b><br/><a NAME="E5:14_3_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">v1</font>,<font color="Maroon">v2</font></span><a href="arytm_0.html#K5">*]</a></span>
 <b>and </b><br/><a NAME="E6:14_3_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">j</font> <a href="xcmplx_0.html#K3" target="_self">*</a> <font color="Maroon">y</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">u1</font>,<font color="Maroon">v1</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">u2</font>,<font color="Maroon">v2</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">u1</font>,<font color="Maroon">v2</font></span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">u2</font>,<font color="Maroon">v1</font></span>)</span></span>)</span></span><a href="arytm_0.html#K5">*]</a></span>
 <b>by </b><i/>;<br/>
<a NAME="E7:14_3_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">u2</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i>, <a class="ref" href="arytm_0.html#T26">ARYTM_0:26</a></i>;<br/>
<a NAME="E8:14_3_1"/><b>then </b>
 <a href="arytm_0.html#K2">*</a> <font color="Maroon">u2</font>,<font color="Maroon">v1</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="E9:14_3_1"/><b>then </b><i><font color="Green">E59</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">u1</font>,<font color="Maroon">v2</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">u2</font>,<font color="Maroon">v1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="arytm_0.html#K2">*</a> <font color="Maroon">u1</font>,<font color="Maroon">v2</font>
 
<b>by </b><i><a class="ref" href="arytm_0.html#T13">ARYTM_0:13</a></i>;<br/>
<a NAME="E10:14_3_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">u1</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i>, , <a class="ref" href="arytm_0.html#D7">ARYTM_0:def 7</a></i>;<br/>
<a NAME="E11:14_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="E12:14_3_1"/><b>then </b><i><font color="Green">E61</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">E62</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">u1</font>,<font color="Maroon">v1</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">u2</font>,<font color="Maroon">v2</font></span>)</span></span>)</span> = 
 <a href="arytm_0.html#K1">+</a> <font color="Maroon">v1</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">u2</font>,<font color="Maroon">v2</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="arytm_0.html#T21">ARYTM_0:21</a></i>
<br/>.= 
 <a href="arytm_0.html#K1">+</a> <font color="Maroon">v1</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">u2</font></span>)</span>,<font color="Maroon">v2</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">v1</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2">*</a> 0,<font color="Maroon">v2</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="arytm_0.html#T26">ARYTM_0:26</a></i>
<br/>.= 
 <a href="arytm_0.html#K1">+</a> <font color="Maroon">v1</font>,0
<b>by </b><i><a class="ref" href="arytm_0.html#T14">ARYTM_0:14</a></i>
<br/>.= 
<font color="Maroon">v1</font>
<b>by </b><i><a class="ref" href="arytm_0.html#T13">ARYTM_0:13</a></i>
;<br/>
<a NAME="E14:14_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">z</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>, </i>;<br/>
<b>consider </b><font color="Maroon">u1</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">u2</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">v1</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">v2</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="E17:14_3_1"/>
<font color="Maroon">x</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">u1</font>,<font color="Maroon">u2</font></span><a href="arytm_0.html#K5">*]</a></span>
 <b>and </b><br/><a NAME="E18:14_3_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">v1</font>,<font color="Maroon">v2</font></span><a href="arytm_0.html#K5">*]</a></span>
 <b>and </b><br/><a NAME="E19:14_3_1"/><i><font color="Green">E64</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K5" target="_self">"</a> </span>)</span> <a href="xcmplx_0.html#K3" target="_self">*</a> <font color="Maroon">z</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">u1</font>,<font color="Maroon">v1</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">u2</font>,<font color="Maroon">v2</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">u1</font>,<font color="Maroon">v2</font></span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">u2</font>,<font color="Maroon">v1</font></span>)</span></span>)</span></span><a href="arytm_0.html#K5">*]</a></span>
 <b>by </b><i/>;<br/>
<a NAME="E20:14_3_1"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">v2</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i>, <a class="ref" href="arytm_0.html#T26">ARYTM_0:26</a></i>;<br/>
<a NAME="E21:14_3_1"/><b>then </b><i><font color="Green">E66</font></i>: 
<font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i>, <a class="ref" href="arytm_0.html#D7">ARYTM_0:def 7</a></i>;<br/>
<b>then </b><i><font color="Green">E67</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">u1</font>,<font color="Maroon">v1</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">u2</font>,<font color="Maroon">v2</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">u2</font>,<font color="Maroon">v2</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">u2</font>,<font color="Maroon">v2</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="ref" href="arytm_0.html#T14">ARYTM_0:14</a></i>
;<br/>
<i><font color="Green">E68</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">u1</font>,<font color="Maroon">v2</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">u2</font>,<font color="Maroon">v1</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">u2</font>,<font color="Maroon">v1</font></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#K2">*</a> <font color="Maroon">u2</font>,<font color="Maroon">v1</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="ref" href="arytm_0.html#T14">ARYTM_0:14</a></i>
;<br/>
<b>assume </b><a NAME="E24:14_3_1"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">x</font> <a href="xcmplx_0.html#K3" target="_self">*</a> <font color="Maroon">y</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">E70</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K5" target="_self">"</a> </span>)</span> <a href="xcmplx_0.html#K3" target="_self">*</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K3" target="_self">*</a> <font color="Maroon">y</font> = 
<span class="p1">(<span class="default"><font color="Maroon">x</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">x</font> <a href="xcmplx_0.html#K3" target="_self">*</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i/>
<br/>.= 
0
<b>by </b><i>, , , , <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">x</font> <a href="xcmplx_0.html#K5" target="_self">"</a> </span>)</span> <a href="xcmplx_0.html#K3" target="_self">*</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K3" target="_self">*</a> <font color="Maroon">y</font> = 
<font color="Maroon">j</font> <a href="xcmplx_0.html#K3" target="_self">*</a> <font color="Maroon">y</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">y</font>
<b>by </b><i>, , , , <a class="ref" href="xcmplx_0.html#D2" target="_self">Def2</a>, <a class="ref" href="arytm_0.html#T21">ARYTM_0:21</a></i>
;<br/>
<b>hence </b><a NAME="E27:14_3_1"/>
contradiction
 <b>by </b><i/>;<br/>


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