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<span class="kw">let </span><font color="Maroon" title="c1">p</font>, <font color="Maroon" title="c2">p1</font>, <font color="Maroon" title="c3">p2</font>, <font color="Maroon" title="c4">p3</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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>) &amp;  not <font color="Maroon" title="c4">p3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>) &amp;  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c3">p2</font>) <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a>  &amp; <font color="Maroon" title="c1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">p2</font> implies  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c3">p2</font>) <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c3">p2</font>) )</span><br/>







<span class="kw">assume </span><a NAME="E1:52"/><span class="lab"><font color="Green" title="E49">A1</font></span>: 
<font color="Maroon" title="c1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>)
 ; <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="c4">p3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>) or  not  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c3">p2</font>) <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a>  or  not <font color="Maroon" title="c1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">p2</font> or  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c3">p2</font>) <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c3">p2</font>) )</span><br/>

<span class="kw">assume </span><a NAME="E2:52"/><span class="lab"><font color="Green" title="E50">A2</font></span>: 
 not <font color="Maroon" title="c4">p3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c3">p2</font>) <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a>  or  not <font color="Maroon" title="c1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">p2</font> or  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c3">p2</font>) <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c3">p2</font>) )</span><br/>

<span class="kw">assume </span><a NAME="E3:52"/><span class="lab"><font color="Green" title="E51">A3</font></span>: 
 <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c3">p2</font>) <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="c3">p2</font> or  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c3">p2</font>) <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c3">p2</font>) )</span><br/>

<span class="kw">assume </span><a NAME="E4:52"/><span class="lab"><font color="Green" title="E52">A4</font></span>: 
<font color="Maroon" title="c1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">p2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c3">p2</font>) <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c3">p2</font>)</span><br/>

<span class="kw">assume </span><a NAME="E5:52"/><span class="lab"><font color="Green" title="E53">A5</font></span>: 
 <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c3">p2</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c3">p2</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:52_1"/>
( <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p1</font> or <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p2</font> or ( <font color="Maroon" title="c2">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">p2</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">p1</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_1"><span class="kw">suppose </span></a><a NAME="E1:52_1_1"/>
<font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:52_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:52"><span class="lab"><font color="Green" title="E53">A5</font></span></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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_2"><span class="kw">suppose </span></a><a NAME="E1:52_1_2"/><span class="lab"><font color="Green" title="E54">A6</font></span>: 
<font color="Maroon" title="c2">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:52_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">p2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:52"><span class="lab"><font color="Green" title="E49">A1</font></span></a>, <a class="ref" href="rltopsp1.html#T70" title="RLTOPSP1:th.70">RLTOPSP1:70</a></span>;<br/><span class="kw">hence </span><a NAME="E3:52_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:52"><span class="lab"><font color="Green" title="E53">A5</font></span></a>, <a class="txt" href="#E1:52_1_2"><span class="lab"><font color="Green" title="E54">A6</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_3"><span class="kw">suppose </span></a><a NAME="E1:52_1_3"/><span class="lab"><font color="Green" title="E54">A7</font></span>: 
( <font color="Maroon" title="c2">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">p2</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">p1</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:52_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A8</font></span>: 
 <a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <font color="Maroon" title="c3">p2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <font color="Maroon" title="c2">p1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_3.html#T4" title="EUCLID_3:th.4">EUCLID_3:4</a></span>;<br/><a NAME="E3:52_1_3"/><span class="lab"><font color="Green" title="E56">A9</font></span>: 
 <a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <font color="Maroon" title="c3">p2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:52"><span class="lab"><font color="Green" title="E52">A4</font></span></a>, <a class="ref" href="euclid_3.html#T4" title="EUCLID_3:th.4">EUCLID_3:4</a></span>;<br/><a NAME="E4:52_1_3"/><span class="lab"><font color="Green" title="E57">A10</font></span>: 
 <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c2">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="c4">p3</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c1">p</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:52"><span class="lab"><font color="Green" title="E49">A1</font></span></a>, <a class="txt" href="#E4:52"><span class="lab"><font color="Green" title="E52">A4</font></span></a>, <a class="ref" href="euclid_6.html#T10" target="_self" title="EUCLID_6:th.10">Th10</a></span>;<br/><a NAME="E5:52_1_3"/><span class="lab"><font color="Green" title="E58">A11</font></span>: 
 <a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <font color="Maroon" title="c2">p1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:52_1_3"><span class="lab"><font color="Green" title="E54">A7</font></span></a>, <a class="ref" href="euclid_3.html#T4" title="EUCLID_3:th.4">EUCLID_3:4</a></span>;<br/><a NAME="E6:52_1_3"/><span class="lab"><font color="Green" title="E59">A12</font></span>: 
 <a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <font color="Maroon" title="c4">p3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:52"><span class="lab"><font color="Green" title="E49">A1</font></span></a>, <a class="txt" href="#E2:52"><span class="lab"><font color="Green" title="E50">A2</font></span></a>, <a class="ref" href="euclid_3.html#T4" title="EUCLID_3:th.4">EUCLID_3:4</a></span>;<br/><a NAME="E7:52_1_3"/><span class="lab"><font color="Green" title="E60">A13</font></span>: 
<font color="Maroon" title="c4">p3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">p1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:52"><span class="lab"><font color="Green" title="E50">A2</font></span></a>, <a class="ref" href="rltopsp1.html#T68" title="RLTOPSP1:th.68">RLTOPSP1:68</a></span>;<br/><a NAME="E8:52_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A14</font></span>: 
 <a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <font color="Maroon" title="c4">p3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <font color="Maroon" title="c2">p1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_3.html#T4" title="EUCLID_3:th.4">EUCLID_3:4</a></span>;<br/><a NAME="E9:52_1_3"/><span class="lab"><font color="Green" title="E62">A15</font></span>: 
<font color="Maroon" title="c4">p3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">p2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:52"><span class="lab"><font color="Green" title="E50">A2</font></span></a>, <a class="ref" href="rltopsp1.html#T68" title="RLTOPSP1:th.68">RLTOPSP1:68</a></span>;<br/><a NAME="E10:52_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A16</font></span>: 
 <a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <font color="Maroon" title="c4">p3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <font color="Maroon" title="c3">p2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_3.html#T4" title="EUCLID_3:th.4">EUCLID_3:4</a></span>;<br/><a NAME="E11:52_1_3"/>
<span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c3">p2</font>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">p3</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c3">p2</font>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c4">p3</font>)</span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_3_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_3_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:52_1_3_1_1"/>
( ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c3">p2</font>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c2">p1</font>)</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">p3</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>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c3">p2</font>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c1">p</font>)</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c4">p3</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>  ) or ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c3">p2</font>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c2">p1</font>)</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">p3</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 5 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c3">p2</font>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c1">p</font>)</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c4">p3</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 5 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a> ) or ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c3">p2</font>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c2">p1</font>)</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">p3</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>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c3">p2</font>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c1">p</font>)</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c4">p3</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 5 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a> ) or ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c3">p2</font>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c2">p1</font>)</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">p3</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 5 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c3">p2</font>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c1">p</font>)</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c4">p3</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><span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:52_1_3"><span class="lab"><font color="Green" title="E63">A16</font></span></a>, <a class="txt" href="#E8:52_1_3"><span class="lab"><font color="Green" title="E61">A14</font></span></a>, <a class="txt" href="#E2:52_1_3"><span class="lab"><font color="Green" title="E55">A8</font></span></a>, <a class="txt" href="#E6:52_1_3"><span class="lab"><font color="Green" title="E59">A12</font></span></a>, <a class="txt" href="#E3:52_1_3"><span class="lab"><font color="Green" title="E56">A9</font></span></a>, <a class="ref" href="complex2.html#T88" title="COMPLEX2:th.88">COMPLEX2:88</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_3_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:52_1_3_1_1_1"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c3">p2</font>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c2">p1</font>)</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">p3</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>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c3">p2</font>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c1">p</font>)</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c4">p3</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>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c3">p2</font>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">p3</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c3">p2</font>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c4">p3</font>)</span>)</span></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:52_1_3_1_1_1"/>
<span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c3">p2</font>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">p3</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c3">p2</font>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c4">p3</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:52_1_3"><span class="lab"><font color="Green" title="E57">A10</font></span></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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_3_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:52_1_3_1_1_2"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c3">p2</font>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c2">p1</font>)</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">p3</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 5 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c3">p2</font>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c1">p</font>)</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c4">p3</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 5 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c3">p2</font>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">p3</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c3">p2</font>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c4">p3</font>)</span>)</span></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:52_1_3_1_1_2"/>
<span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c3">p2</font>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">p3</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c3">p2</font>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c4">p3</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:52_1_3"><span class="lab"><font color="Green" title="E57">A10</font></span></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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_3_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:52_1_3_1_1_3"/><span class="lab"><font color="Green" title="E64">A17</font></span>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c3">p2</font>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c2">p1</font>)</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">p3</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>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c3">p2</font>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c1">p</font>)</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c4">p3</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 5 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c3">p2</font>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">p3</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c3">p2</font>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c4">p3</font>)</span>)</span></span><br/><div class="add"><a NAME="E2:52_1_3_1_1_3"/><span class="lab"><font color="Green" title="E65">A18</font></span>: 
(  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c3">p2</font>) <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">p3</font>) <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="complex2.html#T70" title="COMPLEX2:th.70">COMPLEX2:70</a></span>;<br/><a NAME="E3:52_1_3_1_1_3"/>
 <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c4">p3</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="complex2.html#T70" title="COMPLEX2:th.70">COMPLEX2:70</a></span>;<br/><a NAME="E4:52_1_3_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A19</font></span>: 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c4">p3</font>)</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T24" title="XREAL_1:th.24">XREAL_1:24</a></span>;<br/><a NAME="E5:52_1_3_1_1_3"/>
 <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c3">p2</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="complex2.html#T70" title="COMPLEX2:th.70">COMPLEX2:70</a></span>;<br/><a NAME="E6:52_1_3_1_1_3"/><span class="kw">then </span>
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c3">p2</font>)</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T24" title="XREAL_1:th.24">XREAL_1:24</a></span>;<br/><a NAME="E7:52_1_3_1_1_3"/><span class="kw">then </span>
<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="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c3">p2</font>)</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <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="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c4">p3</font>)</span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:52_1_3_1_1_3"><span class="lab"><font color="Green" title="E66">A19</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E8:52_1_3_1_1_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c3">p2</font>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">p3</font>)</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><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="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c3">p2</font>)</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c4">p3</font>)</span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:52_1_3_1_1_3"><span class="lab"><font color="Green" title="E65">A18</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><span class="kw">hence </span><a NAME="E9:52_1_3_1_1_3"/>
<span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c3">p2</font>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">p3</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c3">p2</font>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c4">p3</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:52_1_3"><span class="lab"><font color="Green" title="E57">A10</font></span></a>, <a class="txt" href="#E1:52_1_3_1_1_3"><span class="lab"><font color="Green" title="E64">A17</font></span></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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_3_1_1_4"><span class="kw">suppose </span></a><a NAME="E1:52_1_3_1_1_4"/><span class="lab"><font color="Green" title="E64">A20</font></span>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c3">p2</font>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c2">p1</font>)</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">p3</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 5 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c3">p2</font>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c1">p</font>)</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c4">p3</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>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c3">p2</font>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">p3</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c3">p2</font>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c4">p3</font>)</span>)</span></span><br/><div class="add"><a NAME="E2:52_1_3_1_1_4"/>
(  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">p3</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a> &amp;  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c3">p2</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="complex2.html#T70" title="COMPLEX2:th.70">COMPLEX2:70</a></span>;<br/><a NAME="E3:52_1_3_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A21</font></span>: 
<span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">p3</font>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c3">p2</font>)</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E4:52_1_3_1_1_4"/>
(  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c3">p2</font>) <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c4">p3</font>) <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="complex2.html#T70" title="COMPLEX2:th.70">COMPLEX2:70</a></span>;<br/><a NAME="E5:52_1_3_1_1_4"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">p3</font>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c3">p2</font>)</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><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="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c3">p2</font>)</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c4">p3</font>)</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:52_1_3_1_1_4"><span class="lab"><font color="Green" title="E65">A21</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><span class="kw">hence </span><a NAME="E6:52_1_3_1_1_4"/>
<span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c3">p2</font>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">p3</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c3">p2</font>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c4">p3</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:52_1_3"><span class="lab"><font color="Green" title="E57">A10</font></span></a>, <a class="txt" href="#E1:52_1_3_1_1_4"><span class="lab"><font color="Green" title="E64">A20</font></span></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/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><a NAME="E12:52_1_3"/><span class="kw">then </span>
 <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">p3</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c4">p3</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:52"><span class="lab"><font color="Green" title="E53">A5</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E13:52_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A22</font></span>: 
 <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">p1</font>,<font color="Maroon" title="c4">p3</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c4">p3</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:52"><span class="lab"><font color="Green" title="E49">A1</font></span></a>, <a class="ref" href="euclid_6.html#T9" target="_self" title="EUCLID_6:th.9">Th9</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_3_2"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:52_1_3_2"/>
( ( <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c4">p3</font>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">p1</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>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">p1</font>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><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">p1</font>,<font color="Maroon" title="c4">p3</font>)</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c1">p</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>  ) or ( <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c4">p3</font>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">p1</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">p1</font>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><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">p1</font>,<font color="Maroon" title="c4">p3</font>)</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c1">p</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>  ) or ( <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c4">p3</font>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">p1</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>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">p1</font>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><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">p1</font>,<font color="Maroon" title="c4">p3</font>)</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c1">p</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 5 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a> ) or ( <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c4">p3</font>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">p1</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">p1</font>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><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">p1</font>,<font color="Maroon" title="c4">p3</font>)</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c1">p</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 5 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:52"><span class="lab"><font color="Green" title="E49">A1</font></span></a>, <a class="txt" href="#E4:52"><span class="lab"><font color="Green" title="E52">A4</font></span></a>, <a class="txt" href="#E8:52_1_3"><span class="lab"><font color="Green" title="E61">A14</font></span></a>, <a class="txt" href="#E6:52_1_3"><span class="lab"><font color="Green" title="E59">A12</font></span></a>, <a class="txt" href="#E5:52_1_3"><span class="lab"><font color="Green" title="E58">A11</font></span></a>, <a class="ref" href="euclid_6.html#T13" target="_self" title="EUCLID_6:th.13">Th13</a>, <a class="ref" href="complex2.html#T88" title="COMPLEX2:th.88">COMPLEX2:88</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_3_2_1"><span class="kw">suppose </span></a><a NAME="E1:52_1_3_2_1"/><span class="lab"><font color="Green" title="E65">A23</font></span>: 
( <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c4">p3</font>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">p1</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>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">p1</font>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><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">p1</font>,<font color="Maroon" title="c4">p3</font>)</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c1">p</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>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:52_1_3_2_1"/>
<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c3">p2</font> <a href="zfmisc_1.html#R1" title="ZFMISC_1:pred.1">are_mutually_different</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:52_1_3"><span class="lab"><font color="Green" title="E54">A7</font></span></a>, <a class="txt" href="#E7:52_1_3"><span class="lab"><font color="Green" title="E60">A13</font></span></a>, <a class="txt" href="#E9:52_1_3"><span class="lab"><font color="Green" title="E62">A15</font></span></a>, <a class="ref" href="zfmisc_1.html#D5" title="ZFMISC_1:def.5">ZFMISC_1:def 5</a></span>;<br/><a NAME="E3:52_1_3_2_1"/><span class="kw">then </span>
 <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">p3</font>) <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</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="#E3:52"><span class="lab"><font color="Green" title="E51">A3</font></span></a>, <a class="ref" href="euclid_6.html#T24" target="_self" title="EUCLID_6:th.24">Th24</a></span>;<br/><a NAME="E4:52_1_3_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A24</font></span>: 
 <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">p1</font>,<font color="Maroon" title="c4">p3</font>) <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</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="#E1:52"><span class="lab"><font color="Green" title="E49">A1</font></span></a>, <a class="txt" href="#E1:52_1_3"><span class="lab"><font color="Green" title="E54">A7</font></span></a>, <a class="ref" href="euclid_6.html#T9" target="_self" title="EUCLID_6:th.9">Th9</a></span>;<br/><a NAME="E5:52_1_3_2_1"/>
<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">p3</font> <a href="zfmisc_1.html#R1" title="ZFMISC_1:pred.1">are_mutually_different</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:52"><span class="lab"><font color="Green" title="E49">A1</font></span></a>, <a class="txt" href="#E2:52"><span class="lab"><font color="Green" title="E50">A2</font></span></a>, <a class="txt" href="#E1:52_1_3"><span class="lab"><font color="Green" title="E54">A7</font></span></a>, <a class="txt" href="#E7:52_1_3"><span class="lab"><font color="Green" title="E60">A13</font></span></a>, <a class="ref" href="zfmisc_1.html#D5" title="ZFMISC_1:def.5">ZFMISC_1:def 5</a></span>;<br/><a NAME="E6:52_1_3_2_1"/><span class="kw">then </span>
(  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c1">p</font>) <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a>  &amp;  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">p1</font>) <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</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="#E4:52_1_3_2_1"><span class="lab"><font color="Green" title="E66">A24</font></span></a>, <a class="ref" href="euclid_6.html#T24" target="_self" title="EUCLID_6:th.24">Th24</a></span>;<br/><a NAME="E7:52_1_3_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">p1</font>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c1">p</font>)</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a> <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="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E8:52_1_3_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A25</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">p1</font>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c1">p</font>)</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><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">p1</font>,<font color="Maroon" title="c4">p3</font>)</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a></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="#E4:52_1_3_2_1"><span class="lab"><font color="Green" title="E66">A24</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E9:52_1_3_2_1"/>
1 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 3 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T68" title="XREAL_1:th.68">XREAL_1:68</a></span>;<br/><span class="kw">hence </span><a NAME="E10:52_1_3_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:52_1_3_2_1"><span class="lab"><font color="Green" title="E65">A23</font></span></a>, <a class="txt" href="#E8:52_1_3_2_1"><span class="lab"><font color="Green" title="E67">A25</font></span></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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_3_2_2"><span class="kw">suppose </span></a><a NAME="E1:52_1_3_2_2"/><span class="lab"><font color="Green" title="E65">A26</font></span>: 
( <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c4">p3</font>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">p1</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">p1</font>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><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">p1</font>,<font color="Maroon" title="c4">p3</font>)</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c1">p</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>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:52_1_3_2_2"/><span class="lab"><font color="Green" title="E66">A27</font></span>: 
(  <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">p1</font>,<font color="Maroon" title="c4">p3</font>) <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c1">p</font>) <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="complex2.html#T70" title="COMPLEX2:th.70">COMPLEX2:70</a></span>;<br/><a NAME="E3:52_1_3_2_2"/>
 <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c4">p3</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><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">p1</font>,<font color="Maroon" title="c4">p3</font>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c1">p</font>)</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:52_1_3_2_2"><span class="lab"><font color="Green" title="E65">A26</font></span></a></span>;<br/><a NAME="E4:52_1_3_2_2"/><span class="kw">then </span>
 <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c4">p3</font>) <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:52_1_3_2_2"><span class="lab"><font color="Green" title="E66">A27</font></span></a>, <a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><span class="kw">hence </span><a NAME="E5:52_1_3_2_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="ref" href="complex2.html#T70" title="COMPLEX2:th.70">COMPLEX2:70</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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_3_2_3"><span class="kw">suppose </span></a><a NAME="E1:52_1_3_2_3"/><span class="lab"><font color="Green" title="E65">A28</font></span>: 
( <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c4">p3</font>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">p1</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>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">p1</font>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><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">p1</font>,<font color="Maroon" title="c4">p3</font>)</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c1">p</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 5 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:52_1_3_2_3"/>
(  <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">p1</font>,<font color="Maroon" title="c4">p3</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a> &amp;  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c1">p</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="complex2.html#T70" title="COMPLEX2:th.70">COMPLEX2:70</a></span>;<br/><a NAME="E3:52_1_3_2_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><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">p1</font>,<font color="Maroon" title="c4">p3</font>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c1">p</font>)</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E4:52_1_3_2_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c4">p3</font>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default">4 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default">4 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:52_1_3_2_3"><span class="lab"><font color="Green" title="E65">A28</font></span></a></span>;<br/><a NAME="E5:52_1_3_2_3"/><span class="kw">then </span>
 <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c4">p3</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><span class="kw">hence </span><a NAME="E6:52_1_3_2_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="ref" href="complex2.html#T70" title="COMPLEX2:th.70">COMPLEX2:70</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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_3_2_4"><span class="kw">suppose </span></a><a NAME="E1:52_1_3_2_4"/>
( <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c4">p3</font>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">p1</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">p1</font>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><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">p1</font>,<font color="Maroon" title="c4">p3</font>)</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c1">p</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 5 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:52_1_3_2_4"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c4">p3</font>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><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">p1</font>,<font color="Maroon" title="c4">p3</font>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c1">p</font>)</span>)</span>
 ;<br/><a NAME="E3:52_1_3_2_4"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c4">p3</font>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c4">p3</font>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c1">p</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:52_1_3"><span class="lab"><font color="Green" title="E64">A22</font></span></a>, <a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><a NAME="E4:52_1_3_2_4"/><span class="kw">then </span>
2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">p3</font>,<font color="Maroon" title="c1">p</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><span class="kw">hence </span><a NAME="E5:52_1_3_2_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="ref" href="complex2.html#T70" title="COMPLEX2:th.70">COMPLEX2:70</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/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
