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<span class="kw">let </span><font color="Maroon" title="c1">b</font>, <font color="Maroon" title="c2">a</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="complex2.html#K3" title="COMPLEX2:func.3">angle</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c1">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies  <a href="complex2.html#K3" title="COMPLEX2:func.3">angle</a> (<font color="Maroon" title="c2">a</font>,<span class="p1">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <font color="Maroon" title="c1">b</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a>  )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:65"/><span class="lab"><font color="Green" title="E54">A1</font></span>: 
<font color="Maroon" title="c1">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">and </span><br/><a NAME="E2:65"/><span class="lab"><font color="Green" title="E55">A2</font></span>: 
 <a href="complex2.html#K3" title="COMPLEX2:func.3">angle</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c1">b</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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="complex2.html#K3" title="COMPLEX2:func.3">angle</a> (<font color="Maroon" title="c2">a</font>,<span class="p1">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <font color="Maroon" title="c1">b</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a> </span><br/>

<a NAME="E3:65"/><span class="lab"><font color="Green" title="E56">A3</font></span>: 
 <a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p1">(<span class="default"><a href="complex2.html#K2" title="COMPLEX2:func.2">Rotate</a> (<font color="Maroon" title="c1">b</font>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p3">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:65"><span class="lab"><font color="Green" title="E54">A1</font></span></a>, <a class="txt" href="#E2:65"><span class="lab"><font color="Green" title="E55">A2</font></span></a>, <a class="ref" href="complex2.html#D3" target="_self" title="COMPLEX2:def.3">Def5</a></span>;<br/>
<a NAME="E4:65"/><span class="lab"><font color="Green" title="E57">A4</font></span>: 
 <a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p1">(<span class="default"><a href="complex2.html#K2" title="COMPLEX2:func.2">Rotate</a> (<span class="p2">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <font color="Maroon" title="c1">b</font></span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p3">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p1">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <span class="p2">(<span class="default"><a href="complex2.html#K2" title="COMPLEX2:func.2">Rotate</a> (<font color="Maroon" title="c1">b</font>,<span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p4">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex2.html#T58" target="_self" title="COMPLEX2:th.58">Th72</a></span>;<br/>
<a NAME="E5:65"/>
 <a href="complex2.html#K2" title="COMPLEX2:func.2">Rotate</a> (<font color="Maroon" title="c1">b</font>,<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:65"><span class="lab"><font color="Green" title="E54">A1</font></span></a>, <a class="ref" href="complex2.html#T52" target="_self" title="COMPLEX2:th.52">Th66</a></span>;<br/>
<span class="kw">then </span> <a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p1">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <span class="p2">(<span class="default"><a href="complex2.html#K2" title="COMPLEX2:func.2">Rotate</a> (<font color="Maroon" title="c1">b</font>,<span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p4">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span>)</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p2">(<span class="default"><a href="complex2.html#K2" title="COMPLEX2:func.2">Rotate</a> (<font color="Maroon" title="c1">b</font>,<span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p4">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span>)</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:65"><span class="lab"><font color="Green" title="E56">A3</font></span></a>, <a class="ref" href="complex2.html#T13" target="_self" title="COMPLEX2:th.13">Th22</a>, <a class="ref" href="comptrig.html#T5" title="COMPTRIG:th.5">COMPTRIG:5</a></span>
<br/>.= 
 <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:65"><span class="lab"><font color="Green" title="E56">A3</font></span></a></span>
;<br/>
<span class="kw">hence </span><a NAME="E7:65"/>
 <a href="complex2.html#K3" title="COMPLEX2:func.3">angle</a> (<font color="Maroon" title="c2">a</font>,<span class="p1">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <font color="Maroon" title="c1">b</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:65"><span class="lab"><font color="Green" title="E54">A1</font></span></a>, <a class="txt" href="#E4:65"><span class="lab"><font color="Green" title="E57">A4</font></span></a>, <a class="ref" href="complex2.html#D3" target="_self" title="COMPLEX2:def.3">Def5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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