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<b>let </b><font color="Maroon" title="c1">p</font>, <font color="Maroon" title="c2">p2</font>, <font color="Maroon" title="c3">p3</font>, <font color="Maroon" title="c4">p1</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c3">p3</font> &amp; <font color="Maroon" title="c1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">p2</font> implies  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c4">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c4">p1</font> )</span><br/>







<b>assume </b><a NAME="E1:27"/>
<font color="Maroon" title="c1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c3">p3</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">p2</font> or  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c4">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c4">p1</font> )</span><br/>

<b>then consider </b><font color="Maroon" title="c5">l</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E3:27"/><i><font color="Green" title="E26">A1</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">l</font> &amp; <font color="Maroon" title="c5">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c5">l</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c2">p2</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">l</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c3">p3</font></span>)</span> )
 <b>by </b><i><a class="ref" href="sppol_1.html#T21" title="SPPOL_1:th.21">SPPOL_1:21</a></i>;<br/>
<i><font color="Green" title="E27">A2</font></i>: <font color="Maroon" title="c1">p</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c5">l</font></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c2">p2</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">l</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c3">p3</font></span>)</span></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font>
<b>by </b><i><a class="txt" href="#E3:27"><i><font color="Green" title="E26">A1</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c2">p2</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c5">l</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c2">p2</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">l</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c3">p3</font></span>)</span></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font>
<b>by </b><i><a class="ref" href="euclid.html#T37" title="EUCLID:th.37">EUCLID:37</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c5">l</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c2">p2</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">l</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c3">p3</font></span>)</span></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font>
<b>by </b><i><a class="ref" href="euclid.html#T33" title="EUCLID:th.33">EUCLID:33</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c5">l</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c2">p2</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">l</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c3">p3</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p2">(<span class="default"><a href="euclid.html#K19" title="EUCLID:func.19">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T30" title="EUCLID:th.30">EUCLID:30</a></i>
<br/>.= 
<font color="Maroon" title="c2">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c5">l</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c2">p2</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">l</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c3">p3</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p3">(<span class="default"><a href="euclid.html#K19" title="EUCLID:func.19">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T30" title="EUCLID:th.30">EUCLID:30</a></i>
<br/>.= 
<font color="Maroon" title="c2">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K19" title="EUCLID:func.19">-</a> <font color="Maroon" title="c2">p2</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c5">l</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c2">p2</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">l</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c3">p3</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T30" title="EUCLID:th.30">EUCLID:30</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p2">(<span class="default"><a href="euclid.html#K19" title="EUCLID:func.19">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c5">l</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c2">p2</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">l</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c3">p3</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T30" title="EUCLID:th.30">EUCLID:30</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> 2</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c5">l</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c2">p2</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">l</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c3">p3</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T40" title="EUCLID:th.40">EUCLID:40</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">l</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c2">p2</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">l</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c3">p3</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T31" title="EUCLID:th.31">EUCLID:31</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">l</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c2">p2</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">l</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c3">p3</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T34" title="EUCLID:th.34">EUCLID:34</a></i>
<br/>.= 
<font color="Maroon" title="c5">l</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T36" title="EUCLID:th.36">EUCLID:36</a></i>
;<br/>
<b>assume </b><a NAME="E5:27"/><i><font color="Green" title="E28">A3</font></i>: 
<font color="Maroon" title="c1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">p2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c4">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c4">p1</font></span><br/>

<b>set </b><font color="Maroon" title="c6">c</font> =  <a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c7">c1</font> =  <a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c8">c3</font> =  <a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span>;<br/>
<a NAME="E6:27"/><i><font color="Green" title="E29">A4</font></i>: 
<font color="Maroon" title="c5">l</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:27_2"/>
<font color="Maroon" title="c5">l</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<b>then </b><font color="Maroon" title="c1">p</font> = 
<span class="p1">(<span class="default">1 <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c2">p2</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> 2</span>)</span>
<b>by </b><i><a class="txt" href="#E3:27"><i><font color="Green" title="E26">A1</font></i></a>, <a class="ref" href="euclid.html#T33" title="EUCLID:th.33">EUCLID:33</a></i>
<br/>.= 
1 <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c2">p2</font>
<b>by </b><i><a class="ref" href="euclid.html#T31" title="EUCLID:th.31">EUCLID:31</a></i>
<br/>.= 
<font color="Maroon" title="c2">p2</font>
<b>by </b><i><a class="ref" href="euclid.html#T33" title="EUCLID:th.33">EUCLID:33</a></i>
;<br/>
<b>hence </b><a NAME="E3:27_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E5:27"><i><font color="Green" title="E28">A3</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
 <a href="euclid_3.html#K1" title="EUCLID_3:func.1">cpx2euc</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">l</font></span>)</span> = 
<font color="Maroon" title="c5">l</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K1" title="EUCLID_3:func.1">cpx2euc</a> <span class="p2">(<span class="default"><a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid_3.html#T23" title="EUCLID_3:th.23">EUCLID_3:23</a></i>
<br/>.= 
<font color="Maroon" title="c5">l</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid_3.html#T2" title="EUCLID_3:th.2">EUCLID_3:2</a></i>
<br/>.= 
 <a href="euclid_3.html#K1" title="EUCLID_3:func.1">cpx2euc</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E4:27"><i><font color="Green" title="E27">A2</font></i></a>, <a class="ref" href="euclid_3.html#T2" title="EUCLID_3:th.2">EUCLID_3:2</a></i>
;<br/>
<a NAME="E8:27"/><b>then </b>
 <a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">l</font>
 
<b>by </b><i><a class="ref" href="euclid_3.html#T5" title="EUCLID_3:th.5">EUCLID_3:5</a></i>;<br/>
<a NAME="E9:27"/><b>then </b><i><font color="Green" title="E30">A5</font></i>: 
 <a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></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="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E3:27"><i><font color="Green" title="E26">A1</font></i></a>, <a class="txt" href="#E6:27"><i><font color="Green" title="E29">A4</font></i></a>, <a class="ref" href="complex2.html#T40" title="COMPLEX2:th.40">COMPLEX2:40</a></i>;<br/>
<a NAME="E10:27"/>
 <a href="complex2.html#K3" title="COMPLEX2:func.3">angle</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">p1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex2.html#K3" title="COMPLEX2:func.3">angle</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">p1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_4"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:27_4_1"/>
(  <a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></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>  or  <a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></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>  or ( not  <a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></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>  &amp; not  <a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></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>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_4_1_1"><b>suppose </b></a><a NAME="E1:27_4_1_1"/><i><font color="Green" title="E31">A6</font></i>: 
(  <a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></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>  or  <a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></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>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="complex2.html#K3" title="COMPLEX2:func.3">angle</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">p1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex2.html#K3" title="COMPLEX2:func.3">angle</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">p1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span>)</span></span><br/><div class="add"><b>then </b> <a href="complex2.html#K3" title="COMPLEX2:func.3">angle</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">p1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span>)</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="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">p1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></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> <span class="p4">(<span class="default"><a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="complex2.html#D5" title="COMPLEX2:def.5">COMPLEX2:def 5</a></i>
<br/>.= 
 <a href="complex2.html#K3" title="COMPLEX2:func.3">angle</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">p1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E9:27"><i><font color="Green" title="E30">A5</font></i></a>, <a class="txt" href="#E1:27_4_1_1"><i><font color="Green" title="E31">A6</font></i></a>, <a class="ref" href="complex2.html#D5" title="COMPLEX2:def.5">COMPLEX2:def 5</a></i>
;<br/><b>hence </b><a NAME="E3:27_4_1_1"/>
 <a href="complex2.html#K3" title="COMPLEX2:func.3">angle</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">p1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex2.html#K3" title="COMPLEX2:func.3">angle</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">p1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_4_1_2"><b>suppose </b></a><a NAME="E1:27_4_1_2"/><i><font color="Green" title="E31">A7</font></i>: 
( not  <a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></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>  &amp; not  <a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></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>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="complex2.html#K3" title="COMPLEX2:func.3">angle</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">p1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex2.html#K3" title="COMPLEX2:func.3">angle</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">p1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span>)</span></span><br/><div class="add"><b>then </b> <a href="complex2.html#K3" title="COMPLEX2:func.3">angle</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">p1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <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="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="complex2.html#D5" title="COMPLEX2:def.5">COMPLEX2:def 5</a></i>
<br/>.= 
 <a href="complex2.html#K3" title="COMPLEX2:func.3">angle</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">p1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E9:27"><i><font color="Green" title="E30">A5</font></i></a>, <a class="txt" href="#E1:27_4_1_2"><i><font color="Green" title="E31">A7</font></i></a>, <a class="ref" href="complex2.html#D5" title="COMPLEX2:def.5">COMPLEX2:def 5</a></i>
;<br/><b>hence </b><a NAME="E3:27_4_1_2"/>
 <a href="complex2.html#K3" title="COMPLEX2:func.3">angle</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">p1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex2.html#K3" title="COMPLEX2:func.3">angle</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">p1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b> <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c4">p1</font> = 
 <a href="complex2.html#K3" title="COMPLEX2:func.3">angle</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">p1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="euclid_6.html#E10"><i><font color="Green" title="E10">Lm7</font></i></a></i>
<br/>.= 
 <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c4">p1</font>
<b>by </b><i><a class="txt" href="euclid_6.html#E10"><i><font color="Green" title="E10">Lm7</font></i></a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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