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

<b>let </b><font color="Maroon">p1</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon">p2</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon">p3</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>





<b>assume </b><a NAME="E1:48"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">p1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p2</font> &amp; <font color="Maroon">p3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p2</font> )
 ;<br/>

<a NAME="E2:48"/><i><font color="Green">E49</font></i>: 
<span class="p1">(<span class="default"><a href="euclid_3.html#K2" target="_self">euc2cpx</a> <font color="Maroon">p1</font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K2" target="_self">euc2cpx</a> <font color="Maroon">p2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="euclid_3.html#K2" target="_self">euc2cpx</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K20">-</a> <font color="Maroon">p2</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E3:48"/><i><font color="Green">E50</font></i>: 
<span class="p1">(<span class="default"><a href="euclid_3.html#K2" target="_self">euc2cpx</a> <font color="Maroon">p3</font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K2" target="_self">euc2cpx</a> <font color="Maroon">p2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="euclid_3.html#K2" target="_self">euc2cpx</a> <span class="p1">(<span class="default"><font color="Maroon">p3</font> <a href="euclid.html#K20">-</a> <font color="Maroon">p2</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E4:48"/><i><font color="Green">E62</font></i>: 
<span class="p1"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K20">-</a> <font color="Maroon">p2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">p3</font> <a href="euclid.html#K20">-</a> <font color="Maroon">p2</font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K3">Re</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_3.html#K2" target="_self">euc2cpx</a> <span class="p3">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K20">-</a> <font color="Maroon">p2</font></span>)</span></span>)</span> <a href="complex2.html#K1">.|.</a> <span class="p2">(<span class="default"><a href="euclid_3.html#K2" target="_self">euc2cpx</a> <span class="p3">(<span class="default"><font color="Maroon">p3</font> <a href="euclid.html#K20">-</a> <font color="Maroon">p2</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E5:48"/>
( <font color="Maroon">p1</font> <a href="euclid.html#K20">-</a> <font color="Maroon">p2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K16">0.REAL</a> 2 &amp; <font color="Maroon">p3</font> <a href="euclid.html#K20">-</a> <font color="Maroon">p2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K16">0.REAL</a> 2 )
 
<b>by </b><i>, <a class="ref" href="euclid.html#T47">EUCLID:47</a></i>;<br/>
<a NAME="E6:48"/><b>then </b><i><font color="Green">E71</font></i>: 
(  <a href="euclid_3.html#K2" target="_self">euc2cpx</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K20">-</a> <font color="Maroon">p2</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="complex1.html#K5">0c</a>  &amp;  <a href="euclid_3.html#K2" target="_self">euc2cpx</a> <span class="p1">(<span class="default"><font color="Maroon">p3</font> <a href="euclid.html#K20">-</a> <font color="Maroon">p2</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="complex1.html#K5">0c</a>  )
 
<b>by </b><i>, </i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:48_1"/>
<span class="p1"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K20">-</a> <font color="Maroon">p2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">p3</font> <a href="euclid.html#K20">-</a> <font color="Maroon">p2</font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:48_1"/><b>then </b>
 <a href="complex1.html#K3">Re</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_3.html#K2" target="_self">euc2cpx</a> <span class="p3">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K20">-</a> <font color="Maroon">p2</font></span>)</span></span>)</span> <a href="complex2.html#K1">.|.</a> <span class="p2">(<span class="default"><a href="euclid_3.html#K2" target="_self">euc2cpx</a> <span class="p3">(<span class="default"><font color="Maroon">p3</font> <a href="euclid.html#K20">-</a> <font color="Maroon">p2</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i/>;<br/><a NAME="E3:48_1"/><b>then </b>
(  <a href="complex2.html#K4">angle</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K2" target="_self">euc2cpx</a> <span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K20">-</a> <font color="Maroon">p2</font></span>)</span></span>)</span>,<a href="complex1.html#K5">0c</a> ,<span class="p1">(<span class="default"><a href="euclid_3.html#K2" target="_self">euc2cpx</a> <span class="p2">(<span class="default"><font color="Maroon">p3</font> <a href="euclid.html#K20">-</a> <font color="Maroon">p2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2 or  <a href="complex2.html#K4">angle</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K2" target="_self">euc2cpx</a> <span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K20">-</a> <font color="Maroon">p2</font></span>)</span></span>)</span>,<a href="complex1.html#K5">0c</a> ,<span class="p1">(<span class="default"><a href="euclid_3.html#K2" target="_self">euc2cpx</a> <span class="p2">(<span class="default"><font color="Maroon">p3</font> <a href="euclid.html#K20">-</a> <font color="Maroon">p2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">3 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a>  )
 <b>by </b><i>, <a class="ref" href="complex2.html#T89">COMPLEX2:89</a></i>;<br/><b>hence </b><a NAME="E4:48_1"/>
(  <a href="euclid_3.html#K4" target="_self">angle</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p3</font> <a href="hidden.html#R1">=</a> <a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2 or  <a href="euclid_3.html#K4" target="_self">angle</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p3</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">3 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a>  )
 <b>by </b><i>, , <a class="ref" href="complex2.html#T85">COMPLEX2:85</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E8:48"/>
(  <a href="euclid_3.html#K4" target="_self">angle</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p3</font> <a href="hidden.html#R1">=</a> <a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2 or  <a href="euclid_3.html#K4" target="_self">angle</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p3</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">3 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a>  )
 ;<br/>

<a NAME="E9:48"/><b>then </b>
(  <a href="complex2.html#K4">angle</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K2" target="_self">euc2cpx</a> <span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K20">-</a> <font color="Maroon">p2</font></span>)</span></span>)</span>,<a href="complex1.html#K5">0c</a> ,<span class="p1">(<span class="default"><a href="euclid_3.html#K2" target="_self">euc2cpx</a> <span class="p2">(<span class="default"><font color="Maroon">p3</font> <a href="euclid.html#K20">-</a> <font color="Maroon">p2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2 or  <a href="complex2.html#K4">angle</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K2" target="_self">euc2cpx</a> <span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K20">-</a> <font color="Maroon">p2</font></span>)</span></span>)</span>,<a href="complex1.html#K5">0c</a> ,<span class="p1">(<span class="default"><a href="euclid_3.html#K2" target="_self">euc2cpx</a> <span class="p2">(<span class="default"><font color="Maroon">p3</font> <a href="euclid.html#K20">-</a> <font color="Maroon">p2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">3 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a>  )
 
<b>by </b><i>, , <a class="ref" href="complex2.html#T85">COMPLEX2:85</a></i>;<br/>
<b>hence </b><a NAME="E10:48"/>
<span class="p1"><a href="euclid_2.html#K2">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K20">-</a> <font color="Maroon">p2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">p3</font> <a href="euclid.html#K20">-</a> <font color="Maroon">p2</font></span>)</span></span><a href="euclid_2.html#K2">)|</a></span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, , <a class="ref" href="complex2.html#T89">COMPLEX2:89</a></i>;<br/>


</div>
