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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></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="E1:49"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font> &amp; (  <a href="euclid_3.html#K4" target="_self">angle</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <a href="sin_cos.html#K33">PI</a>  <a href="real_1.html#K6">/</a> 2 or  <a href="euclid_3.html#K4" target="_self">angle</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></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#K33">PI</a>  ) )
 ;<br/>

<a NAME="E2:49"/><b>then </b><i><font color="Green">E28</font></i>: 
 <a href="euclid_3.html#K2" target="_self">euc2cpx</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid_3.html#K2" target="_self">euc2cpx</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="euclid_3.html#T6" target="_self">Th6</a></i>;<br/>
<a NAME="E3:49"/><i><font color="Green">E29</font></i>: 
 <a href="euclid_3.html#K2" target="_self">euc2cpx</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid_3.html#K2" target="_self">euc2cpx</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="euclid_3.html#E1:49"><i><font color="Green">E27</font></i></a>, <a class="ref" href="euclid_3.html#T6" target="_self">Th6</a></i>;<br/>
<a NAME="E4:49"/><i><font color="Green">E30</font></i>: 
<span class="p1">(<span class="default"><a href="euclid_3.html#K2" target="_self">euc2cpx</a> <font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></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">c<sub>1</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="euclid_3.html#T19" target="_self">Th19</a></i>;<br/>
<a NAME="E5:49"/><i><font color="Green">E31</font></i>: 
<span class="p1">(<span class="default"><a href="euclid_3.html#K2" target="_self">euc2cpx</a> <font color="Maroon">c<sub>3</sub></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">c<sub>2</sub></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">c<sub>3</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="euclid_3.html#T19" target="_self">Th19</a></i>;<br/>
<a NAME="E6:49"/><i><font color="Green">E32</font></i>: 
<span class="p1">(<span class="default"><a href="euclid_3.html#K2" target="_self">euc2cpx</a> <font color="Maroon">c<sub>1</sub></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">c<sub>3</sub></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">c<sub>1</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="euclid_3.html#T19" target="_self">Th19</a></i>;<br/>
<a NAME="E7:49"/><i><font color="Green">E33</font></i>: 
<span class="p1"><a href="complex1.html#K17">|.</a><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">c<sub>1</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>
 
<b>by </b><i><a class="ref" href="euclid_3.html#T31" target="_self">Th31</a></i>;<br/>
<a NAME="E8:49"/><i><font color="Green">E34</font></i>: 
<span class="p1"><a href="complex1.html#K17">|.</a><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">c<sub>3</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>
 
<b>by </b><i><a class="ref" href="euclid_3.html#T31" target="_self">Th31</a></i>;<br/>
<a NAME="E9:49"/>
<span class="p1"><a href="complex1.html#K17">|.</a><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">c<sub>1</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>
 
<b>by </b><i><a class="ref" href="euclid_3.html#T31" target="_self">Th31</a></i>;<br/>
<b>hence </b><a NAME="E10:49"/>
<span class="p1">(<span class="default"><span class="p2"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="square_1.html#K5">^2</a> 
 <b>by </b><i><a class="txt" href="euclid_3.html#E1:49"><i><font color="Green">E27</font></i></a>, <a class="txt" href="euclid_3.html#E2:49"><i><font color="Green">E28</font></i></a>, <a class="txt" href="euclid_3.html#E3:49"><i><font color="Green">E29</font></i></a>, <a class="txt" href="euclid_3.html#E4:49"><i><font color="Green">E30</font></i></a>, <a class="txt" href="euclid_3.html#E5:49"><i><font color="Green">E31</font></i></a>, <a class="txt" href="euclid_3.html#E6:49"><i><font color="Green">E32</font></i></a>, <a class="txt" href="euclid_3.html#E7:49"><i><font color="Green">E33</font></i></a>, <a class="txt" href="euclid_3.html#E8:49"><i><font color="Green">E34</font></i></a>, <a class="ref" href="complex2.html#T91">COMPLEX2:91</a></i>;<br/>


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