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<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="E1:50"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">p2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p1</font> &amp; <font color="Maroon">p1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p3</font> &amp; <font color="Maroon">p3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p2</font> &amp;  <a href="euclid_3.html#K4" target="_self">angle</a> <font color="Maroon">p2</font>,<font color="Maroon">p1</font>,<font color="Maroon">p3</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="sin_cos.html#K35">PI</a>  &amp;  <a href="euclid_3.html#K4" target="_self">angle</a> <font color="Maroon">p1</font>,<font color="Maroon">p3</font>,<font color="Maroon">p2</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="sin_cos.html#K35">PI</a>  &amp;  <a href="euclid_3.html#K4" target="_self">angle</a> <font color="Maroon">p3</font>,<font color="Maroon">p2</font>,<font color="Maroon">p1</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="sin_cos.html#K35">PI</a>  )
 ;<br/>

<a NAME="E2:50"/><b>then </b><i><font color="Green">E49</font></i>: 
 <a href="euclid_3.html#K2" target="_self">euc2cpx</a> <font color="Maroon">p1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid_3.html#K2" target="_self">euc2cpx</a> <font color="Maroon">p2</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:50"/><i><font color="Green">E50</font></i>: 
 <a href="euclid_3.html#K2" target="_self">euc2cpx</a> <font color="Maroon">p1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid_3.html#K2" target="_self">euc2cpx</a> <font color="Maroon">p3</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:50"/><i><font color="Green">E62</font></i>: 
 <a href="euclid_3.html#K2" target="_self">euc2cpx</a> <font color="Maroon">p3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid_3.html#K2" target="_self">euc2cpx</a> <font color="Maroon">p2</font>
 
<b>by </b><i>, </i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:50_1"/>
( 0 <a href="hidden.html#R1">=</a>  <a href="complex2.html#K4">angle</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K2" target="_self">euc2cpx</a> <font color="Maroon">p2</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid_3.html#K2" target="_self">euc2cpx</a> <font color="Maroon">p1</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid_3.html#K2" target="_self">euc2cpx</a> <font color="Maroon">p3</font></span>)</span> or 0 <a href="xxreal_0.html#R1">&lt;</a>  <a href="complex2.html#K4">angle</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K2" target="_self">euc2cpx</a> <font color="Maroon">p2</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid_3.html#K2" target="_self">euc2cpx</a> <font color="Maroon">p1</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid_3.html#K2" target="_self">euc2cpx</a> <font color="Maroon">p3</font></span>)</span> )
 </span><b>by </b><i><a class="ref" href="complex2.html#T84">COMPLEX2:84</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_1"><b>suppose </b></a><a NAME="E1:50_1_1"/><i><font color="Green">E71</font></i>: 
0 <a href="hidden.html#R1">=</a>  <a href="complex2.html#K4">angle</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K2" target="_self">euc2cpx</a> <font color="Maroon">p2</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid_3.html#K2" target="_self">euc2cpx</a> <font color="Maroon">p1</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid_3.html#K2" target="_self">euc2cpx</a> <font color="Maroon">p3</font></span>)</span>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:50_1_1_1_1"/>
( (  <a href="complex2.html#K4">angle</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K2" target="_self">euc2cpx</a> <font color="Maroon">p1</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid_3.html#K2" target="_self">euc2cpx</a> <font color="Maroon">p3</font></span>)</span>,<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> 0 &amp;  <a href="complex2.html#K4">angle</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K2" target="_self">euc2cpx</a> <font color="Maroon">p3</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid_3.html#K2" target="_self">euc2cpx</a> <font color="Maroon">p2</font></span>)</span>,<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="hidden.html#R1">=</a>  <a href="sin_cos.html#K35">PI</a>  ) or (  <a href="complex2.html#K4">angle</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K2" target="_self">euc2cpx</a> <font color="Maroon">p1</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid_3.html#K2" target="_self">euc2cpx</a> <font color="Maroon">p3</font></span>)</span>,<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="sin_cos.html#K35">PI</a>  &amp;  <a href="complex2.html#K4">angle</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K2" target="_self">euc2cpx</a> <font color="Maroon">p3</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid_3.html#K2" target="_self">euc2cpx</a> <font color="Maroon">p2</font></span>)</span>,<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="hidden.html#R1">=</a> 0 ) )
 </span><b>by </b><i>, , , , <a class="ref" href="complex2.html#T101">COMPLEX2:101</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:50_1_1_1_1_1"/>
(  <a href="complex2.html#K4">angle</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K2" target="_self">euc2cpx</a> <font color="Maroon">p1</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid_3.html#K2" target="_self">euc2cpx</a> <font color="Maroon">p3</font></span>)</span>,<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> 0 &amp;  <a href="complex2.html#K4">angle</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K2" target="_self">euc2cpx</a> <font color="Maroon">p3</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid_3.html#K2" target="_self">euc2cpx</a> <font color="Maroon">p2</font></span>)</span>,<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="hidden.html#R1">=</a>  <a href="sin_cos.html#K35">PI</a>  )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:50_1_1_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_3.html#K4" target="_self">angle</a> <font color="Maroon">p2</font>,<font color="Maroon">p1</font>,<font color="Maroon">p3</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="euclid_3.html#K4" target="_self">angle</a> <font color="Maroon">p1</font>,<font color="Maroon">p3</font>,<font color="Maroon">p2</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" target="_self">angle</a> <font color="Maroon">p3</font>,<font color="Maroon">p2</font>,<font color="Maroon">p1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="sin_cos.html#K35">PI</a> 
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:50_1_1_1_1_2"/>
(  <a href="complex2.html#K4">angle</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K2" target="_self">euc2cpx</a> <font color="Maroon">p1</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid_3.html#K2" target="_self">euc2cpx</a> <font color="Maroon">p3</font></span>)</span>,<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="sin_cos.html#K35">PI</a>  &amp;  <a href="complex2.html#K4">angle</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K2" target="_self">euc2cpx</a> <font color="Maroon">p3</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid_3.html#K2" target="_self">euc2cpx</a> <font color="Maroon">p2</font></span>)</span>,<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="hidden.html#R1">=</a> 0 )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:50_1_1_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_3.html#K4" target="_self">angle</a> <font color="Maroon">p2</font>,<font color="Maroon">p1</font>,<font color="Maroon">p3</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="euclid_3.html#K4" target="_self">angle</a> <font color="Maroon">p1</font>,<font color="Maroon">p3</font>,<font color="Maroon">p2</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" target="_self">angle</a> <font color="Maroon">p3</font>,<font color="Maroon">p2</font>,<font color="Maroon">p1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="sin_cos.html#K35">PI</a> 
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:50_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_3.html#K4" target="_self">angle</a> <font color="Maroon">p2</font>,<font color="Maroon">p1</font>,<font color="Maroon">p3</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="euclid_3.html#K4" target="_self">angle</a> <font color="Maroon">p1</font>,<font color="Maroon">p3</font>,<font color="Maroon">p2</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" target="_self">angle</a> <font color="Maroon">p3</font>,<font color="Maroon">p2</font>,<font color="Maroon">p1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="sin_cos.html#K35">PI</a> 
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_2"><b>suppose </b></a><a NAME="E1:50_1_2"/><i><font color="Green">E72</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a>  <a href="complex2.html#K4">angle</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K2" target="_self">euc2cpx</a> <font color="Maroon">p2</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid_3.html#K2" target="_self">euc2cpx</a> <font color="Maroon">p1</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid_3.html#K2" target="_self">euc2cpx</a> <font color="Maroon">p3</font></span>)</span>
 ;<br/><div class="add"><b>thus </b><a NAME="E2:50_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_3.html#K4" target="_self">angle</a> <font color="Maroon">p2</font>,<font color="Maroon">p1</font>,<font color="Maroon">p3</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="euclid_3.html#K4" target="_self">angle</a> <font color="Maroon">p1</font>,<font color="Maroon">p3</font>,<font color="Maroon">p2</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" target="_self">angle</a> <font color="Maroon">p3</font>,<font color="Maroon">p2</font>,<font color="Maroon">p1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="sin_cos.html#K35">PI</a> 
 <b>by </b><i>, , , , <a class="ref" href="complex2.html#T98">COMPLEX2:98</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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