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

<span class="kw">let </span><font color="Maroon" title="c5">c1</font>, <font color="Maroon" title="c6">c2</font> be   <a href="xcmplx_0.html#V1" title="XCMPLX_0:attr.1">complex</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">z</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c2">z</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c6">c2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon" title="c5">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">c2</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:9_1_2"/><span class="lab"><font color="Green" title="E8">A4</font></span>: 
<font color="Maroon" title="c2">z</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">c1</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">and </span><br/><a NAME="E2:9_1_2"/><span class="lab"><font color="Green" title="E9">A5</font></span>: 
<font color="Maroon" title="c2">z</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c6">c2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">c2</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c7">x1</font>, <font color="Maroon" title="c8">x2</font>, <font color="Maroon" title="c9">y1</font>, <font color="Maroon" title="c10">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="E4:9_1_2"/><span class="lab"><font color="Green" title="E10">A6</font></span>: 
<font color="Maroon" title="c2">z</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="c7">x1</font>,<font color="Maroon" title="c8">x2</font></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span>
 <span class="kw">and </span><br/><a NAME="E5:9_1_2"/><span class="lab"><font color="Green" title="E11">A7</font></span>: 
<font color="Maroon" title="c5">c1</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="c9">y1</font>,<font color="Maroon" title="c10">y2</font></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span>
 <span class="kw">and </span><br/><a NAME="E6:9_1_2"/><span class="lab"><font color="Green" title="E12">A8</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</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"><span class="p2">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c7">x1</font>,<font color="Maroon" title="c9">y1</font></span>)</span>,<span class="p2">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c8">x2</font>,<font color="Maroon" title="c10">y2</font></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="txt" href="#E1:9_1_2"><span class="lab"><font color="Green" title="E8">A4</font></span></a>, <a class="ref" href="xcmplx_0.html#D4" target="_self" title="XCMPLX_0:def.4">Def4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c11">x19</font>, <font color="Maroon" title="c12">x29</font>, <font color="Maroon" title="c13">y19</font>, <font color="Maroon" title="c14">y29</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="E8:9_1_2"/><span class="lab"><font color="Green" title="E13">A9</font></span>: 
<font color="Maroon" title="c2">z</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="c11">x19</font>,<font color="Maroon" title="c12">x29</font></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span>
 <span class="kw">and </span><br/><a NAME="E9:9_1_2"/><span class="lab"><font color="Green" title="E14">A10</font></span>: 
<font color="Maroon" title="c6">c2</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="c13">y19</font>,<font color="Maroon" title="c14">y29</font></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span>
 <span class="kw">and </span><br/><a NAME="E10:9_1_2"/><span class="lab"><font color="Green" title="E15">A11</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</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"><span class="p2">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c11">x19</font>,<font color="Maroon" title="c13">y19</font></span>)</span>,<span class="p2">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c12">x29</font>,<font color="Maroon" title="c14">y29</font></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="txt" href="#E2:9_1_2"><span class="lab"><font color="Green" title="E9">A5</font></span></a>, <a class="ref" href="xcmplx_0.html#D4" target="_self" title="XCMPLX_0:def.4">Def4</a></span>;<br/>
<a NAME="E11:9_1_2"/><span class="lab"><font color="Green" title="E16">A12</font></span>: 
<font color="Maroon" title="c7">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x19</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:9_1_2"><span class="lab"><font color="Green" title="E10">A6</font></span></a>, <a class="txt" href="#E8:9_1_2"><span class="lab"><font color="Green" title="E13">A9</font></span></a>, <a class="ref" href="arytm_0.html#T12" title="ARYTM_0:th.12">ARYTM_0:12</a></span>;<br/>
<a NAME="E12:9_1_2"/><span class="lab"><font color="Green" title="E17">A13</font></span>: 
<font color="Maroon" title="c8">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">x29</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:9_1_2"><span class="lab"><font color="Green" title="E10">A6</font></span></a>, <a class="txt" href="#E8:9_1_2"><span class="lab"><font color="Green" title="E13">A9</font></span></a>, <a class="ref" href="arytm_0.html#T12" title="ARYTM_0:th.12">ARYTM_0:12</a></span>;<br/>
<a NAME="E13:9_1_2"/><span class="lab"><font color="Green" title="E18">A14</font></span>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c7">x1</font>,<font color="Maroon" title="c9">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="#E6:9_1_2"><span class="lab"><font color="Green" title="E12">A8</font></span></a>, <a class="txt" href="xcmplx_0.html#E6"><span class="lab"><font color="Green" title="E4">Lm1</font></span></a>, <a class="ref" href="arytm_0.html#T12" title="ARYTM_0:th.12">ARYTM_0:12</a></span>;<br/>
<a NAME="E14:9_1_2"/><span class="lab"><font color="Green" title="E19">A15</font></span>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c7">x1</font>,<font color="Maroon" title="c13">y19</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="#E10:9_1_2"><span class="lab"><font color="Green" title="E15">A11</font></span></a>, <a class="txt" href="#E11:9_1_2"><span class="lab"><font color="Green" title="E16">A12</font></span></a>, <a class="txt" href="xcmplx_0.html#E6"><span class="lab"><font color="Green" title="E4">Lm1</font></span></a>, <a class="ref" href="arytm_0.html#T12" title="ARYTM_0:th.12">ARYTM_0:12</a></span>;<br/>
<a NAME="E15:9_1_2"/><span class="lab"><font color="Green" title="E20">A16</font></span>: 
<font color="Maroon" title="c9">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">y19</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:9_1_2"><span class="lab"><font color="Green" title="E18">A14</font></span></a>, <a class="txt" href="#E14:9_1_2"><span class="lab"><font color="Green" title="E19">A15</font></span></a>, <a class="txt" href="xcmplx_0.html#E8"><span class="lab"><font color="Green" title="E5">Lm2</font></span></a></span>;<br/>
<a NAME="E16:9_1_2"/><span class="lab"><font color="Green" title="E21">A17</font></span>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c8">x2</font>,<font color="Maroon" title="c10">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="#E6:9_1_2"><span class="lab"><font color="Green" title="E12">A8</font></span></a>, <a class="txt" href="xcmplx_0.html#E6"><span class="lab"><font color="Green" title="E4">Lm1</font></span></a>, <a class="ref" href="arytm_0.html#T12" title="ARYTM_0:th.12">ARYTM_0:12</a></span>;<br/>
<a NAME="E17:9_1_2"/><span class="lab"><font color="Green" title="E22">A18</font></span>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c8">x2</font>,<font color="Maroon" title="c14">y29</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="#E10:9_1_2"><span class="lab"><font color="Green" title="E15">A11</font></span></a>, <a class="txt" href="#E12:9_1_2"><span class="lab"><font color="Green" title="E17">A13</font></span></a>, <a class="txt" href="xcmplx_0.html#E6"><span class="lab"><font color="Green" title="E4">Lm1</font></span></a>, <a class="ref" href="arytm_0.html#T12" title="ARYTM_0:th.12">ARYTM_0:12</a></span>;<br/>
<span class="kw">thus </span><a NAME="E18:9_1_2"/>
<font color="Maroon" title="c5">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">c2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:9_1_2"><span class="lab"><font color="Green" title="E11">A7</font></span></a>, <a class="txt" href="#E9:9_1_2"><span class="lab"><font color="Green" title="E14">A10</font></span></a>, <a class="txt" href="#E15:9_1_2"><span class="lab"><font color="Green" title="E20">A16</font></span></a>, <a class="txt" href="#E16:9_1_2"><span class="lab"><font color="Green" title="E21">A17</font></span></a>, <a class="txt" href="#E17:9_1_2"><span class="lab"><font color="Green" title="E22">A18</font></span></a>, <a class="txt" href="xcmplx_0.html#E8"><span class="lab"><font color="Green" title="E5">Lm2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
