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

<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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/>

<b>assume </b><b>that </b><br/><a NAME="E1:9_1_2"/><i><font color="Green" title="E7">A2</font></i>: 
<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> 
 <b>and </b><br/><a NAME="E2:9_1_2"/><i><font color="Green" title="E8">A3</font></i>: 
<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:()"><i><font color="Red">::  thesis: </font></i></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/>

<b>consider </b><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> <b> such that </b><br/><a NAME="E4:9_1_2"/><i><font color="Green" title="E9">A4</font></i>: 
<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>
 <b>and </b><br/><a NAME="E5:9_1_2"/><i><font color="Green" title="E10">A5</font></i>: 
<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>
 <b>and </b><br/><a NAME="E6:9_1_2"/><i><font color="Green" title="E11">A6</font></i>: 
 <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>
 <b>by </b><i><a class="txt" href="xcmplx_0.html#E1:9_1_2"><i><font color="Green" title="E7">A2</font></i></a>, <a class="ref" href="xcmplx_0.html#D4" target="_self" title="XCMPLX_0:def.4">Def4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c11">x1'</font>, <font color="Maroon" title="c12">x2'</font>, <font color="Maroon" title="c13">y1'</font>, <font color="Maroon" title="c14">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="E8:9_1_2"/><i><font color="Green" title="E12">A7</font></i>: 
<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">x1'</font>,<font color="Maroon" title="c12">x2'</font></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span>
 <b>and </b><br/><a NAME="E9:9_1_2"/><i><font color="Green" title="E13">A8</font></i>: 
<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">y1'</font>,<font color="Maroon" title="c14">y2'</font></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span>
 <b>and </b><br/><a NAME="E10:9_1_2"/><i><font color="Green" title="E14">A9</font></i>: 
 <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">x1'</font>,<font color="Maroon" title="c13">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="c12">x2'</font>,<font color="Maroon" title="c14">y2'</font></span>)</span></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span>
 <b>by </b><i><a class="txt" href="xcmplx_0.html#E2:9_1_2"><i><font color="Green" title="E8">A3</font></i></a>, <a class="ref" href="xcmplx_0.html#D4" target="_self" title="XCMPLX_0:def.4">Def4</a></i>;<br/>
<a NAME="E11:9_1_2"/><i><font color="Green" title="E15">A10</font></i>: 
( <font color="Maroon" title="c7">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x1'</font> &amp; <font color="Maroon" title="c8">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">x2'</font> )
 
<b>by </b><i><a class="txt" href="xcmplx_0.html#E4:9_1_2"><i><font color="Green" title="E9">A4</font></i></a>, <a class="txt" href="xcmplx_0.html#E8:9_1_2"><i><font color="Green" title="E12">A7</font></i></a>, <a class="ref" href="arytm_0.html#T12" title="ARYTM_0:th.12">ARYTM_0:12</a></i>;<br/>
<a NAME="E12:9_1_2"/><i><font color="Green" title="E16">A11</font></i>: 
 <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> 
 
<b>by </b><i><a class="txt" href="xcmplx_0.html#E6:9_1_2"><i><font color="Green" title="E11">A6</font></i></a>, <a class="txt" href="xcmplx_0.html#E6"><i><font color="Green" title="E4">Lm1</font></i></a>, <a class="ref" href="arytm_0.html#T12" title="ARYTM_0:th.12">ARYTM_0:12</a></i>;<br/>
<a NAME="E13:9_1_2"/>
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c7">x1</font>,<font color="Maroon" title="c13">y1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="xcmplx_0.html#E10:9_1_2"><i><font color="Green" title="E14">A9</font></i></a>, <a class="txt" href="xcmplx_0.html#E11:9_1_2"><i><font color="Green" title="E15">A10</font></i></a>, <a class="txt" href="xcmplx_0.html#E6"><i><font color="Green" title="E4">Lm1</font></i></a>, <a class="ref" href="arytm_0.html#T12" title="ARYTM_0:th.12">ARYTM_0:12</a></i>;<br/>
<a NAME="E14:9_1_2"/><b>then </b><i><font color="Green" title="E17">A12</font></i>: 
<font color="Maroon" title="c9">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">y1'</font>
 
<b>by </b><i><a class="txt" href="xcmplx_0.html#E12:9_1_2"><i><font color="Green" title="E16">A11</font></i></a>, <a class="txt" href="xcmplx_0.html#E8"><i><font color="Green" title="E5">Lm2</font></i></a></i>;<br/>
<a NAME="E15:9_1_2"/><i><font color="Green" title="E18">A13</font></i>: 
 <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> 
 
<b>by </b><i><a class="txt" href="xcmplx_0.html#E6:9_1_2"><i><font color="Green" title="E11">A6</font></i></a>, <a class="txt" href="xcmplx_0.html#E6"><i><font color="Green" title="E4">Lm1</font></i></a>, <a class="ref" href="arytm_0.html#T12" title="ARYTM_0:th.12">ARYTM_0:12</a></i>;<br/>
<a NAME="E16:9_1_2"/>
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <font color="Maroon" title="c8">x2</font>,<font color="Maroon" title="c14">y2'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="xcmplx_0.html#E10:9_1_2"><i><font color="Green" title="E14">A9</font></i></a>, <a class="txt" href="xcmplx_0.html#E11:9_1_2"><i><font color="Green" title="E15">A10</font></i></a>, <a class="txt" href="xcmplx_0.html#E6"><i><font color="Green" title="E4">Lm1</font></i></a>, <a class="ref" href="arytm_0.html#T12" title="ARYTM_0:th.12">ARYTM_0:12</a></i>;<br/>
<b>hence </b><a NAME="E17:9_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>
 <b>by </b><i><a class="txt" href="xcmplx_0.html#E5:9_1_2"><i><font color="Green" title="E10">A5</font></i></a>, <a class="txt" href="xcmplx_0.html#E9:9_1_2"><i><font color="Green" title="E13">A8</font></i></a>, <a class="txt" href="xcmplx_0.html#E14:9_1_2"><i><font color="Green" title="E17">A12</font></i></a>, <a class="txt" href="xcmplx_0.html#E15:9_1_2"><i><font color="Green" title="E18">A13</font></i></a>, <a class="txt" href="xcmplx_0.html#E8"><i><font color="Green" title="E5">Lm2</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
