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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="E1:46"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">p1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K16">0.REAL</a> 2 &amp; <font color="Maroon">p2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K16">0.REAL</a> 2 )
 ;<br/>

<a NAME="E2:46"/><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="complex1.html#K5">0c</a> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E3:46"/><i><font color="Green">E50</font></i>: 
 <a href="euclid_3.html#K2" target="_self">euc2cpx</a> <font color="Maroon">p2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="complex1.html#K5">0c</a> 
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E4:46"/>
<span class="p1"><a href="euclid_2.html#K2">|(</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></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> <font color="Maroon">p1</font></span>)</span> <a href="complex2.html#K1">.|.</a> <span class="p2">(<span class="default"><a href="euclid_3.html#K2" target="_self">euc2cpx</a> <font color="Maroon">p2</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:46"/>
( <span class="p1"><a href="euclid_2.html#K2">|(</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="euclid_2.html#K2">)|</a></span> <a href="hidden.html#R1">=</a> 0 iff (  <a href="euclid_3.html#K4" target="_self">angle</a> <font color="Maroon">p1</font>,<span class="p1">(<span class="default"><a href="euclid.html#K16">0.REAL</a> 2</span>)</span>,<font color="Maroon">p2</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>,<span class="p1">(<span class="default"><a href="euclid.html#K16">0.REAL</a> 2</span>)</span>,<font color="Maroon">p2</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#T89">COMPLEX2:89</a></i>;<br/>


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