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<b>assume </b><a NAME="E1:14_2_1"/><i><font color="Green" title="E10">A9</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xcmplx_0.html#D3" target="_self" title="XCMPLX_0:def.3">XCMPLX_0:def 3</a><br/>

<a NAME="E2:14_2_1"/>
<font color="Maroon" title="c2">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="ref" href="xcmplx_0.html#D7" target="_self" title="XCMPLX_0:def.7">Def7</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c3">x1</font>, <font color="Maroon" title="c4">x2</font>, <font color="Maroon" title="c5">y1</font>, <font color="Maroon" title="c6">y2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <b> such that </b><br/><a NAME="E4:14_2_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c4">x2</font></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span>
 <b>and </b><br/><a NAME="E5:14_2_1"/><i><font color="Green" title="E11">A10</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c6">y2</font></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span>
 <b>and </b><br/><a NAME="E6:14_2_1"/><i><font color="Green" title="E12">A11</font></i>: 
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><span class="p2">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c5">y1</font></span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c6">y2</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c6">y2</font></span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">y1</font></span>)</span></span>)</span></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span>
 <b>by </b><i><a class="ref" href="xcmplx_0.html#D5" target="_self" title="XCMPLX_0:def.5">Def5</a></i>;<br/>
<a NAME="E7:14_2_1"/><i><font color="Green" title="E13">A12</font></i>: 
<font color="Maroon" title="c6">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 
<b>by </b><i><a class="txt" href="xcmplx_0.html#E1:14_2_1"><i><font color="Green" title="E10">A9</font></i></a>, <a class="txt" href="xcmplx_0.html#E5:14_2_1"><i><font color="Green" title="E11">A10</font></i></a>, <a class="ref" href="arytm_0.html#T26" title="ARYTM_0:th.26">ARYTM_0:26</a></i>;<br/>
<a NAME="E8:14_2_1"/><b>then </b><i><font color="Green" title="E14">A13</font></i>: 
<font color="Maroon" title="c5">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 
<b>by </b><i><a class="txt" href="xcmplx_0.html#E1:14_2_1"><i><font color="Green" title="E10">A9</font></i></a>, <a class="txt" href="xcmplx_0.html#E5:14_2_1"><i><font color="Green" title="E11">A10</font></i></a>, <a class="ref" href="arytm_0.html#D7" title="ARYTM_0:def.7">ARYTM_0:def 7</a></i>;<br/>
<a NAME="E9:14_2_1"/>
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c6">y2</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">y1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 
<b>by </b><i><a class="txt" href="xcmplx_0.html#E6:14_2_1"><i><font color="Green" title="E12">A11</font></i></a>, <a class="ref" href="arytm_0.html#T26" title="ARYTM_0:th.26">ARYTM_0:26</a></i>;<br/>
<b>then </b>1 = 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c5">y1</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <span class="p2">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c6">y2</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="xcmplx_0.html#E6:14_2_1"><i><font color="Green" title="E12">A11</font></i></a>, <a class="ref" href="arytm_0.html#D7" title="ARYTM_0:def.7">ARYTM_0:def 7</a></i>
<br/>.= 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c5">y1</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <span class="p2">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">x2</font></span>)</span>,<font color="Maroon" title="c6">y2</font></span>)</span>
<b>by </b><i><a class="ref" href="arytm_0.html#T17" title="ARYTM_0:th.17">ARYTM_0:17</a></i>
<br/>.= 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> 0,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <span class="p2">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">x2</font></span>)</span>,<font color="Maroon" title="c6">y2</font></span>)</span>
<b>by </b><i><a class="txt" href="xcmplx_0.html#E8:14_2_1"><i><font color="Green" title="E14">A13</font></i></a>, <a class="ref" href="arytm_0.html#T14" title="ARYTM_0:th.14">ARYTM_0:14</a></i>
<br/>.= 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">x2</font></span>)</span>,<font color="Maroon" title="c6">y2</font>
<b>by </b><i><a class="ref" href="arytm_0.html#T13" title="ARYTM_0:th.13">ARYTM_0:13</a></i>
;<br/>
<b>hence </b><a NAME="E11:14_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="xcmplx_0.html#E7:14_2_1"><i><font color="Green" title="E13">A12</font></i></a>, <a class="ref" href="arytm_0.html#T14" title="ARYTM_0:th.14">ARYTM_0:14</a></i>;<br/>


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