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<b>let </b><font color="Maroon" title="c1">p1</font>, <font color="Maroon" title="c2">p2</font>, <font color="Maroon" title="c3">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="euclid_6.html#K1" title="EUCLID_6:func.1">the_area_of_polygon3</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c3">p3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">p1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K21" title="SIN_COS:func.21">sin</a> <span class="p3">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c1">p1</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span><br/>





<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:23_1"/>
( <font color="Maroon" title="c1">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p2</font> or <font color="Maroon" title="c1">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">p2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1"><b>suppose </b></a><a NAME="E1:23_1_1"/><i><font color="Green" title="E22">A1</font></i>: 
<font color="Maroon" title="c1">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="euclid_6.html#K1" title="EUCLID_6:func.1">the_area_of_polygon3</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c3">p3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">p1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K21" title="SIN_COS:func.21">sin</a> <span class="p3">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c1">p1</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span><br/><div class="add"><b>then </b><span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">p1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> = 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> 2</span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T46" title="EUCLID:th.46">EUCLID:46</a></i>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="ref" href="euclid_2.html#T61" title="EUCLID_2:th.61">EUCLID_2:61</a></i>
;<br/><b>hence </b><a NAME="E3:23_1_1"/>
 <a href="euclid_6.html#K1" title="EUCLID_6:func.1">the_area_of_polygon3</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c3">p3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">p1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K21" title="SIN_COS:func.21">sin</a> <span class="p3">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c1">p1</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 <b>by </b><i><a class="txt" href="euclid_6.html#E1:23_1_1"><i><font color="Green" title="E22">A1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_2"><b>suppose </b></a><a NAME="E1:23_1_2"/><i><font color="Green" title="E22">A2</font></i>: 
<font color="Maroon" title="c1">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">p2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="euclid_6.html#K1" title="EUCLID_6:func.1">the_area_of_polygon3</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c3">p3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">p1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K21" title="SIN_COS:func.21">sin</a> <span class="p3">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c1">p1</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span><br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:23_1_2_1"/>
( <font color="Maroon" title="c2">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p3</font> or <font color="Maroon" title="c2">p2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">p3</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_2_1_1"><b>suppose </b></a><a NAME="E1:23_1_2_1_1"/><i><font color="Green" title="E23">A3</font></i>: 
<font color="Maroon" title="c2">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p3</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="euclid_6.html#K1" title="EUCLID_6:func.1">the_area_of_polygon3</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c3">p3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">p1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K21" title="SIN_COS:func.21">sin</a> <span class="p3">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c1">p1</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span><br/><div class="add"><b>then </b><span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> = 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> 2</span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T46" title="EUCLID:th.46">EUCLID:46</a></i>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="ref" href="euclid_2.html#T61" title="EUCLID_2:th.61">EUCLID_2:61</a></i>
;<br/><b>hence </b><a NAME="E3:23_1_2_1_1"/>
 <a href="euclid_6.html#K1" title="EUCLID_6:func.1">the_area_of_polygon3</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c3">p3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">p1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K21" title="SIN_COS:func.21">sin</a> <span class="p3">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c1">p1</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 <b>by </b><i><a class="txt" href="euclid_6.html#E1:23_1_2_1_1"><i><font color="Green" title="E23">A3</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_2_1_2"><b>suppose </b></a><a NAME="E1:23_1_2_1_2"/><i><font color="Green" title="E23">A4</font></i>: 
<font color="Maroon" title="c2">p2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">p3</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="euclid_6.html#K1" title="EUCLID_6:func.1">the_area_of_polygon3</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c3">p3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">p1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K21" title="SIN_COS:func.21">sin</a> <span class="p3">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c1">p1</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span><br/><div class="add"><b>set </b><font color="Maroon" title="c4">a</font> =  <a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span>;<br/><b>set </b><font color="Maroon" title="c5">b</font> =  <a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span>;<br/><div><i><font color="Green" title="E24">A5</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_2_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:23_1_2_1_2_1"/><i><font color="Green" title="E25">A6</font></i>: 
(  <a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  <a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_2_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:23_1_2_1_2_1_1"/>
(  <a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  <a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span><b>by </b><i><a class="txt" href="euclid_6.html#E1:23_1_2_1_2_1"><i><font color="Green" title="E25">A6</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_2_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:23_1_2_1_2_1_1_1"/>
 <a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><b>hence </b><a NAME="E2:23_1_2_1_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="euclid_6.html#E1:23_1_2"><i><font color="Green" title="E22">A2</font></i></a>, <a class="ref" href="euclid.html#T47" title="EUCLID:th.47">EUCLID:47</a>, <a class="ref" href="euclid_3.html#T22" title="EUCLID_3:th.22">EUCLID_3:22</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_2_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:23_1_2_1_2_1_1_2"/>
 <a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><b>hence </b><a NAME="E2:23_1_2_1_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="euclid_6.html#E1:23_1_2_1_2"><i><font color="Green" title="E23">A4</font></i></a>, <a class="ref" href="euclid.html#T47" title="EUCLID:th.47">EUCLID:47</a>, <a class="ref" href="euclid_3.html#T22" title="EUCLID_3:th.22">EUCLID_3:22</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><i><font color="Green" title="E25">A7</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_2_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:23_1_2_1_2_2"/><i><font color="Green" title="E26">A8</font></i>: 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">p1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_2_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:23_1_2_1_2_2_1"/>
( <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">p1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span><b>by </b><i><a class="txt" href="euclid_6.html#E1:23_1_2_1_2_2"><i><font color="Green" title="E26">A8</font></i></a>, <a class="ref" href="xcmplx_1.html#T6" title="XCMPLX_1:th.6">XCMPLX_1:6</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_2_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:23_1_2_1_2_2_1_1"/>
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">p1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><b>hence </b><a NAME="E2:23_1_2_1_2_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="euclid_6.html#E2:23_1_2_1_2"><i><font color="Green" title="E24">A5</font></i></a>, <a class="ref" href="complex1.html#T131" title="COMPLEX1:th.131">COMPLEX1:131</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_2_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:23_1_2_1_2_2_1_2"/>
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><b>hence </b><a NAME="E2:23_1_2_1_2_2_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="euclid_6.html#E2:23_1_2_1_2"><i><font color="Green" title="E24">A5</font></i></a>, <a class="ref" href="complex1.html#T131" title="COMPLEX1:th.131">COMPLEX1:131</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">p1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K21" title="SIN_COS:func.21">sin</a> <span class="p3">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c1">p1</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">p1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <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="sin_cos.html#K21" title="SIN_COS:func.21">sin</a> <span class="p4">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c3">p3</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
<b>by </b><i><a class="ref" href="euclid_6.html#T2" target="_self" title="EUCLID_6:th.2">Th2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">p1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <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="sin_cos.html#K21" title="SIN_COS:func.21">sin</a> <span class="p4">(<span class="default"><a href="complex2.html#K3" title="COMPLEX2:func.3">angle</a> <span class="p5">(<span class="default"><a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">p1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
<b>by </b><i><a class="txt" href="euclid_6.html#E10"><i><font color="Green" title="E10">Lm7</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">p1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <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="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">p1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span>)</span> <a href="complex2.html#K1" title="COMPLEX2:func.1">.|.</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p5">(<span class="default"><span class="p0"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p1">(<span class="default"><a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">p1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p0"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p1">(<span class="default"><a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
<b>by </b><i><a class="txt" href="euclid_6.html#E2:23_1_2_1_2"><i><font color="Green" title="E24">A5</font></i></a>, <a class="ref" href="complex2.html#T83" title="COMPLEX2:th.83">COMPLEX2:83</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p4">(<span class="default"><a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">p1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">p1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span>)</span> <a href="complex2.html#K1" title="COMPLEX2:func.1">.|.</a> <span class="p5">(<span class="default"><a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default"><span class="p4"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p5">(<span class="default"><a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">p1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p5">(<span class="default"><a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
<b>by </b><i><a class="txt" href="euclid_6.html#E20"><i><font color="Green" title="E20">Lm16</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p4">(<span class="default"><a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">p1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p4">(<span class="default"><a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">p1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span>)</span> <a href="complex2.html#K1" title="COMPLEX2:func.1">.|.</a> <span class="p5">(<span class="default"><a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default"><span class="p4"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p5">(<span class="default"><a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">p1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p5">(<span class="default"><a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
<b>by </b><i><a class="txt" href="euclid_6.html#E20"><i><font color="Green" title="E20">Lm16</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">p1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span>)</span> <a href="complex2.html#K1" title="COMPLEX2:func.1">.|.</a> <span class="p4">(<span class="default"><a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p5">(<span class="default"><a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">p1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p5">(<span class="default"><a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default"><span class="p4"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p5">(<span class="default"><a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">p1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p5">(<span class="default"><a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
<b>by </b><i><a class="ref" href="xcmplx_1.html#T82" title="XCMPLX_1:th.82">XCMPLX_1:82</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">p1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span>)</span> <a href="complex2.html#K1" title="COMPLEX2:func.1">.|.</a> <span class="p4">(<span class="default"><a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 1</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
<b>by </b><i><a class="txt" href="euclid_6.html#E3:23_1_2_1_2"><i><font color="Green" title="E25">A7</font></i></a>, <a class="ref" href="xcmplx_1.html#T60" title="XCMPLX_1:th.60">XCMPLX_1:60</a></i>
<br/>.= 
<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"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
<b>by </b><i><a class="txt" href="euclid_6.html#E20"><i><font color="Green" title="E20">Lm16</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c2">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c2">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2

;<br/><b>hence </b><a NAME="E5:23_1_2_1_2"/>
 <a href="euclid_6.html#K1" title="EUCLID_6:func.1">the_area_of_polygon3</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c3">p3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">p1</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K21" title="SIN_COS:func.21">sin</a> <span class="p3">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c1">p1</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div>
