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<span class="kw">assume </span><a NAME="E1:15_2_1"/><span class="lab"><font color="Green" title="E10">A8</font></span>: 
<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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xcmplx_0.html#D3" target="_self" title="XCMPLX_0:def.3">XCMPLX_0:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:15_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
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_0.html#D7" target="_self" title="XCMPLX_0:def.7">Def7</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><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> <span class="kw"> such that </span><br/><a NAME="E4:15_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>
 <span class="kw">and </span><br/><a NAME="E5:15_2_1"/><span class="lab"><font color="Green" title="E11">A9</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E6:15_2_1"/><span class="lab"><font color="Green" title="E12">A10</font></span>: 
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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_0.html#D5" target="_self" title="XCMPLX_0:def.5">Def5</a></span>;<br/>
<a NAME="E7:15_2_1"/>
<font color="Maroon" title="c6">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15_2_1"><span class="lab"><font color="Green" title="E10">A8</font></span></a>, <a class="txt" href="#E5:15_2_1"><span class="lab"><font color="Green" title="E11">A9</font></span></a>, <a class="ref" href="arytm_0.html#T24" title="ARYTM_0:th.24">ARYTM_0:24</a></span>;<br/>
<a NAME="E8:15_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E13">A11</font></span>: 
<font color="Maroon" title="c5">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15_2_1"><span class="lab"><font color="Green" title="E10">A8</font></span></a>, <a class="txt" href="#E5:15_2_1"><span class="lab"><font color="Green" title="E11">A9</font></span></a>, <a class="ref" href="arytm_0.html#D5" title="ARYTM_0:def.5">ARYTM_0:def 5</a></span>;<br/>
<a NAME="E9:15_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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:15_2_1"><span class="lab"><font color="Green" title="E12">A10</font></span></a>, <a class="ref" href="arytm_0.html#T24" title="ARYTM_0:th.24">ARYTM_0:24</a></span>;<br/>
<span class="kw">then </span>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>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:15_2_1"><span class="lab"><font color="Green" title="E12">A10</font></span></a>, <a class="ref" href="arytm_0.html#D5" title="ARYTM_0:def.5">ARYTM_0:def 5</a></span>
<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>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_0.html#T15" title="ARYTM_0:th.15">ARYTM_0:15</a></span>
<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>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:15_2_1"><span class="lab"><font color="Green" title="E13">A11</font></span></a>, <a class="ref" href="arytm_0.html#T11" title="ARYTM_0:th.11">ARYTM_0:11</a>, <a class="ref" href="arytm_0.html#T12" title="ARYTM_0:th.12">ARYTM_0:12</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E11:15_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15_2_1"><span class="lab"><font color="Green" title="E10">A8</font></span></a>, <a class="txt" href="#E5:15_2_1"><span class="lab"><font color="Green" title="E11">A9</font></span></a>, <a class="ref" href="arytm_0.html#T12" title="ARYTM_0:th.12">ARYTM_0:12</a>, <a class="ref" href="arytm_0.html#T24" title="ARYTM_0:th.24">ARYTM_0:24</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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