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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">p</font>, <font color="Maroon" title="c4">pc</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"> for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">r</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Maroon" title="c1">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jgraph_6.html#K5" title="JGRAPH_6:func.5">circle</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">r</font> &amp; <font color="Maroon" title="c2">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jgraph_6.html#K5" title="JGRAPH_6:func.5">circle</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">r</font> &amp; <font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jgraph_6.html#K5" title="JGRAPH_6:func.5">circle</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">r</font> &amp; <font color="Maroon" title="c4">pc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c1">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">p</font> &amp; <font color="Maroon" title="c2">p2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">p</font> &amp; not 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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">p1</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c2">p2</font> holds <br/>2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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">p1</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c2">p2</font></span><br/><b>let </b><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">b</font>, <font color="Maroon" title="c7">r</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jgraph_6.html#K5" title="JGRAPH_6:func.5">circle</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">r</font> &amp; <font color="Maroon" title="c2">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jgraph_6.html#K5" title="JGRAPH_6:func.5">circle</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">r</font> &amp; <font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jgraph_6.html#K5" title="JGRAPH_6:func.5">circle</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">r</font> &amp; <font color="Maroon" title="c4">pc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c1">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">p</font> &amp; <font color="Maroon" title="c2">p2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">p</font> &amp; not 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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">p1</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c2">p2</font> implies 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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">p1</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c2">p2</font> )</span><br/>













<b>assume </b><a NAME="E1:57"/><i><font color="Green" title="E54">A1</font></i>: 
<font color="Maroon" title="c1">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jgraph_6.html#K5" title="JGRAPH_6:func.5">circle</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">r</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c2">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jgraph_6.html#K5" title="JGRAPH_6:func.5">circle</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">r</font> or  not <font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jgraph_6.html#K5" title="JGRAPH_6:func.5">circle</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">r</font> or  not <font color="Maroon" title="c4">pc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> or  not <font color="Maroon" title="c1">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">p</font> or  not <font color="Maroon" title="c2">p2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">p</font> or 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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">p1</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c2">p2</font> or 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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">p1</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c2">p2</font> )</span><br/>

<b>assume </b><a NAME="E2:57"/><i><font color="Green" title="E55">A2</font></i>: 
<font color="Maroon" title="c2">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jgraph_6.html#K5" title="JGRAPH_6:func.5">circle</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">r</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jgraph_6.html#K5" title="JGRAPH_6:func.5">circle</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">r</font> or  not <font color="Maroon" title="c4">pc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> or  not <font color="Maroon" title="c1">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">p</font> or  not <font color="Maroon" title="c2">p2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">p</font> or 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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">p1</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c2">p2</font> or 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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">p1</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c2">p2</font> )</span><br/>

<b>assume </b><a NAME="E3:57"/><i><font color="Green" title="E56">A3</font></i>: 
<font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jgraph_6.html#K5" title="JGRAPH_6:func.5">circle</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">r</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c4">pc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> or  not <font color="Maroon" title="c1">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">p</font> or  not <font color="Maroon" title="c2">p2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">p</font> or 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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">p1</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c2">p2</font> or 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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">p1</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c2">p2</font> )</span><br/>

<b>assume </b><a NAME="E4:57"/><i><font color="Green" title="E57">A4</font></i>: 
<font color="Maroon" title="c4">pc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c1">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">p</font> or  not <font color="Maroon" title="c2">p2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">p</font> or 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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">p1</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c2">p2</font> or 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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">p1</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c2">p2</font> )</span><br/>

<b>assume </b><a NAME="E5:57"/><i><font color="Green" title="E58">A5</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="c3">p</font> &amp; <font color="Maroon" title="c2">p2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">p</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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">p1</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c2">p2</font> or 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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">p1</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c2">p2</font> )</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:57_1"/>
( <font color="Maroon" title="c7">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c7">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_1_1"><b>suppose </b></a><a NAME="E1:57_1_1"/>
<font color="Maroon" title="c7">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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">p1</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c2">p2</font> or 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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">p1</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c2">p2</font> )</span><br/><div class="add"><a NAME="E2:57_1_1"/><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="c4">pc</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <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="c2">p2</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c4">pc</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <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">p</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c4">pc</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="euclid_6.html#E1:57"><i><font color="Green" title="E54">A1</font></i></a>, <a class="txt" href="euclid_6.html#E2:57"><i><font color="Green" title="E55">A2</font></i></a>, <a class="txt" href="euclid_6.html#E3:57"><i><font color="Green" title="E56">A3</font></i></a>, <a class="txt" href="euclid_6.html#E4:57"><i><font color="Green" title="E57">A4</font></i></a>, <a class="ref" href="topreal9.html#T43" title="TOPREAL9:th.43">TOPREAL9:43</a></i>;<br/><a NAME="E3:57_1_1"/><b>then </b><i><font color="Green" title="E59">A6</font></i>: 
( <font color="Maroon" title="c1">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">pc</font> &amp; <font color="Maroon" title="c2">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">pc</font> &amp; <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">pc</font> )
 <b>by </b><i><a class="txt" href="euclid_6.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a></i>;<br/><b>then </b>2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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">p1</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span> = 
2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="ref" href="complex2.html#T93" title="COMPLEX2:th.93">COMPLEX2:93</a></i>
<br/>.= 
 <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c4">pc</font>
<b>by </b><i><a class="ref" href="complex2.html#T93" title="COMPLEX2:th.93">COMPLEX2:93</a></i>
;<br/><b>hence </b><a NAME="E5:57_1_1"/>
( 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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">p1</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c2">p2</font> or 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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">p1</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c2">p2</font> )
 <b>by </b><i><a class="txt" href="euclid_6.html#E3:57_1_1"><i><font color="Green" title="E59">A6</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="57_1_2"><b>suppose </b></a><a NAME="E1:57_1_2"/><i><font color="Green" title="E59">A7</font></i>: 
<font color="Maroon" title="c7">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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">p1</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c2">p2</font> or 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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">p1</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c2">p2</font> )</span><br/><div class="add"><a NAME="E2:57_1_2"/><i><font color="Green" title="E60">A8</font></i>: 
( <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="c4">pc</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r</font> &amp; <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="c2">p2</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c4">pc</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r</font> &amp; <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">p</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c4">pc</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r</font> )
 <b>by </b><i><a class="txt" href="euclid_6.html#E1:57"><i><font color="Green" title="E54">A1</font></i></a>, <a class="txt" href="euclid_6.html#E2:57"><i><font color="Green" title="E55">A2</font></i></a>, <a class="txt" href="euclid_6.html#E3:57"><i><font color="Green" title="E56">A3</font></i></a>, <a class="txt" href="euclid_6.html#E4:57"><i><font color="Green" title="E57">A4</font></i></a>, <a class="ref" href="topreal9.html#T43" title="TOPREAL9:th.43">TOPREAL9:43</a></i>;<br/><a NAME="E3:57_1_2"/>
<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="c4">pc</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><a NAME="E4:57_1_2"/><b>then </b>
<font color="Maroon" title="c7">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="euclid_6.html#E1:57"><i><font color="Green" title="E54">A1</font></i></a>, <a class="txt" href="euclid_6.html#E4:57"><i><font color="Green" title="E57">A4</font></i></a>, <a class="txt" href="euclid_6.html#E1:57_1_2"><i><font color="Green" title="E59">A7</font></i></a>, <a class="ref" href="topreal9.html#T43" title="TOPREAL9:th.43">TOPREAL9:43</a></i>;<br/><b>then consider </b><font color="Maroon" title="c8">p3</font> being   <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><b> such that </b><br/><a NAME="E6:57_1_2"/><i><font color="Green" title="E61">A9</font></i>: 
( <font color="Maroon" title="c3">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">p3</font> &amp; <font color="Maroon" title="c8">p3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jgraph_6.html#K5" title="JGRAPH_6:func.5">circle</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">r</font> &amp; <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c8">p3</font> )
 <b>by </b><i><a class="txt" href="euclid_6.html#E3:57"><i><font color="Green" title="E56">A3</font></i></a>, <a class="ref" href="euclid_6.html#T32" target="_self" title="EUCLID_6:th.32">Th32</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:57_1_2_1"/>
( <font color="Maroon" title="c2">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">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="c8">p3</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_1_2_1_1"><b>suppose </b></a><a NAME="E1:57_1_2_1_1"/>
<font color="Maroon" title="c2">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">p3</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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">p1</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c2">p2</font> or 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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">p1</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c2">p2</font> )</span><br/><div class="add"><b>hence </b><a NAME="E2:57_1_2_1_1"/>
( 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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">p1</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c2">p2</font> or 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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">p1</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c2">p2</font> )
 <b>by </b><i><a class="txt" href="euclid_6.html#E1:57"><i><font color="Green" title="E54">A1</font></i></a>, <a class="txt" href="euclid_6.html#E3:57"><i><font color="Green" title="E56">A3</font></i></a>, <a class="txt" href="euclid_6.html#E4:57"><i><font color="Green" title="E57">A4</font></i></a>, <a class="txt" href="euclid_6.html#E5:57"><i><font color="Green" title="E58">A5</font></i></a>, <a class="txt" href="euclid_6.html#E6:57_1_2"><i><font color="Green" title="E61">A9</font></i></a>, <a class="ref" href="euclid_6.html#T31" target="_self" title="EUCLID_6:th.31">Th31</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="57_1_2_1_2"><b>suppose </b></a><a NAME="E1:57_1_2_1_2"/><i><font color="Green" title="E62">A10</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="c8">p3</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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">p1</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c2">p2</font> or 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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">p1</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c2">p2</font> )</span><br/><div class="add"><a NAME="E2:57_1_2_1_2"/><i><font color="Green" title="E63">A11</font></i>: 
 <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c8">p3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1_2_1_2_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon" title="c9">z2</font> =  <a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p2</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c4">pc</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c10">z3</font> =  <a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">p3</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c4">pc</font></span>)</span>;<br/>
<b>assume </b><a NAME="E1:57_1_2_1_2_1"/>
 <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c8">p3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:57_1_2_1_2_1"/><b>then </b><i><font color="Green" title="E64">A12</font></i>: 
 <a href="euclid_3.html#K3" title="EUCLID_3:func.3">Arg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p2</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c4">pc</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K3" title="EUCLID_3:func.3">Arg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">p3</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c4">pc</font></span>)</span>
 
<b>by </b><i><a class="ref" href="euclid_3.html#T45" title="EUCLID_3:th.45">EUCLID_3:45</a></i>;<br/>
<a NAME="E3:57_1_2_1_2_1"/><i><font color="Green" title="E65">A13</font></i>: 
<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="c2">p2</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c4">pc</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c8">p3</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c4">pc</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
 
<b>by </b><i><a class="txt" href="euclid_6.html#E4:57"><i><font color="Green" title="E57">A4</font></i></a>, <a class="txt" href="euclid_6.html#E2:57_1_2"><i><font color="Green" title="E60">A8</font></i></a>, <a class="txt" href="euclid_6.html#E6:57_1_2"><i><font color="Green" title="E61">A9</font></i></a>, <a class="ref" href="topreal9.html#T43" title="TOPREAL9:th.43">TOPREAL9:43</a></i>;<br/>
<i><font color="Green" title="E66">A14</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="c2">p2</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c4">pc</font></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> = 
<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="c2">p2</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c4">pc</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<b>by </b><i><a class="ref" href="euclid_3.html#T31" title="EUCLID_3:th.31">EUCLID_3:31</a></i>
<br/>.= 
<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="c8">p3</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c4">pc</font></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
<b>by </b><i><a class="txt" href="euclid_6.html#E3:57_1_2_1_2_1"><i><font color="Green" title="E65">A13</font></i></a>, <a class="ref" href="euclid_3.html#T31" title="EUCLID_3:th.31">EUCLID_3:31</a></i>
;<br/>
<i><font color="Green" title="E67">A15</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="c2">p2</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c4">pc</font></span>)</span> = 
<span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><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="c2">p2</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c4">pc</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">(<span class="default"><a href="sin_cos.html#K24" title="SIN_COS:func.24">cos</a> <span class="p4">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</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="c2">p2</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c4">pc</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>,<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="c2">p2</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c4">pc</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">(<span class="default"><a href="sin_cos.html#K21" title="SIN_COS:func.21">sin</a> <span class="p4">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</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="c2">p2</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c4">pc</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span>
<b>by </b><i><a class="ref" href="complex2.html#T19" title="COMPLEX2:th.19">COMPLEX2:19</a></i>
<br/>.= 
 <a href="euclid_3.html#K2" title="EUCLID_3:func.2">euc2cpx</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">p3</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c4">pc</font></span>)</span>
<b>by </b><i><a class="txt" href="euclid_6.html#E2:57_1_2_1_2_1"><i><font color="Green" title="E64">A12</font></i></a>, <a class="txt" href="euclid_6.html#E4:57_1_2_1_2_1"><i><font color="Green" title="E66">A14</font></i></a>, <a class="ref" href="complex2.html#T19" title="COMPLEX2:th.19">COMPLEX2:19</a></i>
;<br/>
<font color="Maroon" title="c2">p2</font> = 
<font color="Maroon" title="c2">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> 2</span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T31" title="EUCLID:th.31">EUCLID:31</a></i>
<br/>.= 
<font color="Maroon" title="c2">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">pc</font> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p2">(<span class="default"><a href="euclid.html#K19" title="EUCLID:func.19">-</a> <font color="Maroon" title="c4">pc</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T40" title="EUCLID:th.40">EUCLID:40</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p2">(<span class="default"><a href="euclid.html#K19" title="EUCLID:func.19">-</a> <font color="Maroon" title="c4">pc</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <font color="Maroon" title="c4">pc</font>
<b>by </b><i><a class="ref" href="euclid.html#T30" title="EUCLID:th.30">EUCLID:30</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">p3</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c4">pc</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <font color="Maroon" title="c4">pc</font>
<b>by </b><i><a class="txt" href="euclid_6.html#E5:57_1_2_1_2_1"><i><font color="Green" title="E67">A15</font></i></a>, <a class="ref" href="euclid_3.html#T6" title="EUCLID_3:th.6">EUCLID_3:6</a></i>
<br/>.= 
<font color="Maroon" title="c8">p3</font> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">pc</font> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p2">(<span class="default"><a href="euclid.html#K19" title="EUCLID:func.19">-</a> <font color="Maroon" title="c4">pc</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T30" title="EUCLID:th.30">EUCLID:30</a></i>
<br/>.= 
<font color="Maroon" title="c8">p3</font> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> 2</span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T40" title="EUCLID:th.40">EUCLID:40</a></i>
<br/>.= 
<font color="Maroon" title="c8">p3</font>
<b>by </b><i><a class="ref" href="euclid.html#T31" title="EUCLID:th.31">EUCLID:31</a></i>
;<br/>
<b>hence </b><a NAME="E7:57_1_2_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="euclid_6.html#E1:57_1_2_1_2"><i><font color="Green" title="E62">A10</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E3:57_1_2_1_2"/><i><font color="Green" title="E64">A16</font></i>: 
( 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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">p2</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c8">p3</font></span>)</span> <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="c2">p2</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c8">p3</font> or 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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="c2">p2</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c8">p3</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <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="c2">p2</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c8">p3</font> )
 <b>by </b><i><a class="txt" href="euclid_6.html#E2:57"><i><font color="Green" title="E55">A2</font></i></a>, <a class="txt" href="euclid_6.html#E3:57"><i><font color="Green" title="E56">A3</font></i></a>, <a class="txt" href="euclid_6.html#E4:57"><i><font color="Green" title="E57">A4</font></i></a>, <a class="txt" href="euclid_6.html#E5:57"><i><font color="Green" title="E58">A5</font></i></a>, <a class="txt" href="euclid_6.html#E6:57_1_2"><i><font color="Green" title="E61">A9</font></i></a>, <a class="ref" href="euclid_6.html#T31" target="_self" title="EUCLID_6:th.31">Th31</a></i>;<br/><a NAME="E4:57_1_2_1_2"/><i><font color="Green" title="E65">A17</font></i>: 
 <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c8">p3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1_2_1_2_2"><b>proof </b></a><div class="add">

<a NAME="E1:57_1_2_1_2_2"/><i><font color="Green" title="E66">A18</font></i>: 
 <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#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
;<br/>
<b>assume </b><a NAME="E2:57_1_2_1_2_2"/>
 <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c8">p3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<b>hence </b><a NAME="E3:57_1_2_1_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="euclid_6.html#E2:57_1_2_1_2"><i><font color="Green" title="E63">A11</font></i></a>, <a class="txt" href="euclid_6.html#E3:57_1_2_1_2"><i><font color="Green" title="E64">A16</font></i></a>, <a class="txt" href="euclid_6.html#E1:57_1_2_1_2_2"><i><font color="Green" title="E66">A18</font></i></a>, <a class="ref" href="complex2.html#T84" title="COMPLEX2:th.84">COMPLEX2:84</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E5:57_1_2_1_2"/><b>then </b><i><font color="Green" title="E66">A19</font></i>: 
 <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c8">p3</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.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#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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">p2</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c8">p3</font></span>)</span>
 <b>by </b><i><a class="ref" href="euclid_3.html#T46" title="EUCLID_3:th.46">EUCLID_3:46</a></i>;<br/><a NAME="E6:57_1_2_1_2"/><i><font color="Green" title="E67">A20</font></i>: 
( 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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">p2</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c8">p3</font></span>)</span> <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="c2">p2</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c8">p3</font> implies 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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="c8">p3</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <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="c8">p3</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c2">p2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1_2_1_2_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:57_1_2_1_2_3"/>
2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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">p2</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c8">p3</font></span>)</span> <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="c2">p2</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c8">p3</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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="c8">p3</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <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="c8">p3</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c2">p2</font></span><br/>

<b>hence </b>2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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="c8">p3</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> = 
<span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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">p2</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c8">p3</font></span>)</span>
<b>by </b><i><a class="txt" href="euclid_6.html#E5:57_1_2_1_2"><i><font color="Green" title="E66">A19</font></i></a></i>
<br/>.= 
 <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c8">p3</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c2">p2</font>
<b>by </b><i><a class="txt" href="euclid_6.html#E2:57_1_2_1_2"><i><font color="Green" title="E63">A11</font></i></a>, <a class="ref" href="euclid_3.html#T46" title="EUCLID_3:th.46">EUCLID_3:46</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E7:57_1_2_1_2"/><i><font color="Green" title="E68">A21</font></i>: 
( 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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="c2">p2</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c8">p3</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <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="c2">p2</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c8">p3</font> implies 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c8">p3</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span> <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="c8">p3</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c2">p2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1_2_1_2_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:57_1_2_1_2_4"/><i><font color="Green" title="E69">A22</font></i>: 
2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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="c2">p2</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c8">p3</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <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="c2">p2</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c8">p3</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c8">p3</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span> <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="c8">p3</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c2">p2</font></span><br/>

<b>thus </b>2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c8">p3</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span> = 
2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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#K35" title="SIN_COS:func.35">PI</a> </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="c2">p2</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c8">p3</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="euclid_6.html#E4:57_1_2_1_2"><i><font color="Green" title="E65">A17</font></i></a>, <a class="ref" href="euclid_3.html#T46" title="EUCLID_3:th.46">EUCLID_3:46</a></i>
<br/>.= 
<span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default">2 <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="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c8">p3</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span></span>)</span>

<br/>.= 
 <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c8">p3</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c2">p2</font>
<b>by </b><i><a class="txt" href="euclid_6.html#E2:57_1_2_1_2"><i><font color="Green" title="E63">A11</font></i></a>, <a class="txt" href="euclid_6.html#E1:57_1_2_1_2_4"><i><font color="Green" title="E69">A22</font></i></a>, <a class="ref" href="euclid_3.html#T46" title="EUCLID_3:th.46">EUCLID_3:46</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E8:57_1_2_1_2"/><i><font color="Green" title="E69">A23</font></i>: 
(  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</font> <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">p1</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c8">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="c8">p3</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span> or <span class="p1">(<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="c3">p</font>,<font color="Maroon" title="c2">p2</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#K35" title="SIN_COS:func.35">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">p1</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c8">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="c8">p3</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span> )
 <b>by </b><i><a class="ref" href="euclid_6.html#T4" target="_self" title="EUCLID_6:th.4">Th4</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_1_2_1_2_5"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:57_1_2_1_2_5"/>
(  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c2">p2</font> <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">p1</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c8">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="c8">p3</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c2">p2</font></span>)</span> or <span class="p1">(<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="c4">pc</font>,<font color="Maroon" title="c2">p2</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#K35" title="SIN_COS:func.35">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">p1</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c8">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="c8">p3</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c2">p2</font></span>)</span> )
 </span><b>by </b><i><a class="ref" href="euclid_6.html#T4" target="_self" title="EUCLID_6:th.4">Th4</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_1_2_1_2_5_1"><b>suppose </b></a><a NAME="E1:57_1_2_1_2_5_1"/><i><font color="Green" title="E70">A24</font></i>: 
 <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c2">p2</font> <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">p1</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c8">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="c8">p3</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c2">p2</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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">p1</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c2">p2</font> or 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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">p1</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c2">p2</font> )</span><br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_1_2_1_2_5_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:57_1_2_1_2_5_1_1"/>
( ( 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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">p1</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c8">p3</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c8">p3</font> &amp; 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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="c8">p3</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <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="c8">p3</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c2">p2</font> ) or ( 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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">p1</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c8">p3</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c8">p3</font> &amp; 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c8">p3</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span> <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="c8">p3</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c2">p2</font> ) or ( 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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">p1</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c8">p3</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c8">p3</font> &amp; 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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="c8">p3</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <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="c8">p3</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c2">p2</font> ) or ( 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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">p1</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c8">p3</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c8">p3</font> &amp; 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c8">p3</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span> <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="c8">p3</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c2">p2</font> ) )
 </span><b>by </b><i><a class="txt" href="euclid_6.html#E1:57"><i><font color="Green" title="E54">A1</font></i></a>, <a class="txt" href="euclid_6.html#E2:57"><i><font color="Green" title="E55">A2</font></i></a>, <a class="txt" href="euclid_6.html#E3:57"><i><font color="Green" title="E56">A3</font></i></a>, <a class="txt" href="euclid_6.html#E4:57"><i><font color="Green" title="E57">A4</font></i></a>, <a class="txt" href="euclid_6.html#E5:57"><i><font color="Green" title="E58">A5</font></i></a>, <a class="txt" href="euclid_6.html#E6:57_1_2"><i><font color="Green" title="E61">A9</font></i></a>, <a class="txt" href="euclid_6.html#E6:57_1_2_1_2"><i><font color="Green" title="E67">A20</font></i></a>, <a class="txt" href="euclid_6.html#E7:57_1_2_1_2"><i><font color="Green" title="E68">A21</font></i></a>, <a class="ref" href="euclid_6.html#T31" target="_self" title="EUCLID_6:th.31">Th31</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_1_2_1_2_5_1_1_1"><b>suppose </b></a><a NAME="E1:57_1_2_1_2_5_1_1_1"/>
( 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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">p1</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c8">p3</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c8">p3</font> &amp; 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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="c8">p3</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <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="c8">p3</font>,<font color="Maroon" title="c4">pc</font>,<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"> ( 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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">p1</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c2">p2</font> or 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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">p1</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c2">p2</font> )</span><br/><div class="add"><a NAME="E2:57_1_2_1_2_5_1_1_1"/><b>then </b>
(  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c2">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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">p1</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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#K35" title="SIN_COS:func.35">PI</a> </span>)</span> or  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c2">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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">p1</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</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#K35" title="SIN_COS:func.35">PI</a> </span>)</span> )
 <b>by </b><i><a class="txt" href="euclid_6.html#E8:57_1_2_1_2"><i><font color="Green" title="E69">A23</font></i></a>, <a class="txt" href="euclid_6.html#E1:57_1_2_1_2_5_1"><i><font color="Green" title="E70">A24</font></i></a></i>;<br/><b>hence </b><a NAME="E3:57_1_2_1_2_5_1_1_1"/>
( 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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">p1</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c2">p2</font> or 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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">p1</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c2">p2</font> )
 <b>by </b><i><a class="txt" href="euclid_6.html#E6"><i><font color="Green" title="E6">Lm3</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="57_1_2_1_2_5_1_1_2"><b>suppose </b></a><a NAME="E1:57_1_2_1_2_5_1_1_2"/>
( 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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">p1</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c8">p3</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c8">p3</font> &amp; 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c8">p3</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span> <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="c8">p3</font>,<font color="Maroon" title="c4">pc</font>,<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"> ( 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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">p1</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c2">p2</font> or 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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">p1</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c2">p2</font> )</span><br/><div class="add"><a NAME="E2:57_1_2_1_2_5_1_1_2"/><b>then </b>
(  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c2">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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">p1</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span> or  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c2">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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">p1</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span></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#K35" title="SIN_COS:func.35">PI</a> </span>)</span> )
 <b>by </b><i><a class="txt" href="euclid_6.html#E8:57_1_2_1_2"><i><font color="Green" title="E69">A23</font></i></a>, <a class="txt" href="euclid_6.html#E1:57_1_2_1_2_5_1"><i><font color="Green" title="E70">A24</font></i></a></i>;<br/><b>hence </b><a NAME="E3:57_1_2_1_2_5_1_1_2"/>
( 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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">p1</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c2">p2</font> or 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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">p1</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c2">p2</font> )
 <b>by </b><i><a class="txt" href="euclid_6.html#E7"><i><font color="Green" title="E7">Lm4</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="57_1_2_1_2_5_1_1_3"><b>suppose </b></a><a NAME="E1:57_1_2_1_2_5_1_1_3"/>
( 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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">p1</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c8">p3</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c8">p3</font> &amp; 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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="c8">p3</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <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="c8">p3</font>,<font color="Maroon" title="c4">pc</font>,<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"> ( 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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">p1</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c2">p2</font> or 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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">p1</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c2">p2</font> )</span><br/><div class="add"><a NAME="E2:57_1_2_1_2_5_1_1_3"/><b>then </b>
 <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c2">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <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="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c8">p3</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c8">p3</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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#K35" title="SIN_COS:func.35">PI</a> </span>)</span>
 <b>by </b><i><a class="txt" href="euclid_6.html#E1:57_1_2_1_2_5_1"><i><font color="Green" title="E70">A24</font></i></a></i>;<br/><b>hence </b><a NAME="E3:57_1_2_1_2_5_1_1_3"/>
( 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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">p1</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c2">p2</font> or 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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">p1</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c2">p2</font> )
 <b>by </b><i><a class="txt" href="euclid_6.html#E8:57_1_2_1_2"><i><font color="Green" title="E69">A23</font></i></a>, <a class="txt" href="euclid_6.html#E8"><i><font color="Green" title="E8">Lm5</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="57_1_2_1_2_5_1_1_4"><b>suppose </b></a><a NAME="E1:57_1_2_1_2_5_1_1_4"/>
( 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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">p1</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c8">p3</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c8">p3</font> &amp; 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c8">p3</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span> <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="c8">p3</font>,<font color="Maroon" title="c4">pc</font>,<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"> ( 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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">p1</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c2">p2</font> or 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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">p1</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c2">p2</font> )</span><br/><div class="add"><a NAME="E2:57_1_2_1_2_5_1_1_4"/><b>then </b>
(  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c2">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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">p1</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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#K35" title="SIN_COS:func.35">PI</a> </span>)</span> or  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c2">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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">p1</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</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#K35" title="SIN_COS:func.35">PI</a> </span>)</span> )
 <b>by </b><i><a class="txt" href="euclid_6.html#E8:57_1_2_1_2"><i><font color="Green" title="E69">A23</font></i></a>, <a class="txt" href="euclid_6.html#E1:57_1_2_1_2_5_1"><i><font color="Green" title="E70">A24</font></i></a></i>;<br/><b>hence </b><a NAME="E3:57_1_2_1_2_5_1_1_4"/>
( 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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">p1</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c2">p2</font> or 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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">p1</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c2">p2</font> )
 <b>by </b><i><a class="txt" href="euclid_6.html#E6"><i><font color="Green" title="E6">Lm3</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><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_1_2_1_2_5_2"><b>suppose </b></a><a NAME="E1:57_1_2_1_2_5_2"/><i><font color="Green" title="E70">A25</font></i>: 
<span class="p1">(<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="c4">pc</font>,<font color="Maroon" title="c2">p2</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#K35" title="SIN_COS:func.35">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">p1</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c8">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="c8">p3</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c2">p2</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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">p1</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c2">p2</font> or 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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">p1</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c2">p2</font> )</span><br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_1_2_1_2_5_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:57_1_2_1_2_5_2_1"/>
( ( 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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">p1</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c8">p3</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c8">p3</font> &amp; 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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="c8">p3</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <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="c8">p3</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c2">p2</font> ) or ( 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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">p1</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c8">p3</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c8">p3</font> &amp; 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c8">p3</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span> <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="c8">p3</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c2">p2</font> ) or ( 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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">p1</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c8">p3</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c8">p3</font> &amp; 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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="c8">p3</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <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="c8">p3</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c2">p2</font> ) or ( 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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">p1</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c8">p3</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c8">p3</font> &amp; 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c8">p3</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span> <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="c8">p3</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c2">p2</font> ) )
 </span><b>by </b><i><a class="txt" href="euclid_6.html#E1:57"><i><font color="Green" title="E54">A1</font></i></a>, <a class="txt" href="euclid_6.html#E2:57"><i><font color="Green" title="E55">A2</font></i></a>, <a class="txt" href="euclid_6.html#E3:57"><i><font color="Green" title="E56">A3</font></i></a>, <a class="txt" href="euclid_6.html#E4:57"><i><font color="Green" title="E57">A4</font></i></a>, <a class="txt" href="euclid_6.html#E5:57"><i><font color="Green" title="E58">A5</font></i></a>, <a class="txt" href="euclid_6.html#E6:57_1_2"><i><font color="Green" title="E61">A9</font></i></a>, <a class="txt" href="euclid_6.html#E6:57_1_2_1_2"><i><font color="Green" title="E67">A20</font></i></a>, <a class="txt" href="euclid_6.html#E7:57_1_2_1_2"><i><font color="Green" title="E68">A21</font></i></a>, <a class="ref" href="euclid_6.html#T31" target="_self" title="EUCLID_6:th.31">Th31</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_1_2_1_2_5_2_1_1"><b>suppose </b></a><a NAME="E1:57_1_2_1_2_5_2_1_1"/>
( 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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">p1</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c8">p3</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c8">p3</font> &amp; 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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="c8">p3</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <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="c8">p3</font>,<font color="Maroon" title="c4">pc</font>,<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"> ( 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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">p1</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c2">p2</font> or 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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">p1</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c2">p2</font> )</span><br/><div class="add"><a NAME="E2:57_1_2_1_2_5_2_1_1"/><b>then </b>
 <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c2">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <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="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c8">p3</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c8">p3</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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#K35" title="SIN_COS:func.35">PI</a> </span>)</span>
 <b>by </b><i><a class="txt" href="euclid_6.html#E1:57_1_2_1_2_5_2"><i><font color="Green" title="E70">A25</font></i></a></i>;<br/><b>hence </b><a NAME="E3:57_1_2_1_2_5_2_1_1"/>
( 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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">p1</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c2">p2</font> or 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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">p1</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c2">p2</font> )
 <b>by </b><i><a class="txt" href="euclid_6.html#E8:57_1_2_1_2"><i><font color="Green" title="E69">A23</font></i></a>, <a class="txt" href="euclid_6.html#E8"><i><font color="Green" title="E8">Lm5</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="57_1_2_1_2_5_2_1_2"><b>suppose </b></a><a NAME="E1:57_1_2_1_2_5_2_1_2"/>
( 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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">p1</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c8">p3</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c8">p3</font> &amp; 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c8">p3</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span> <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="c8">p3</font>,<font color="Maroon" title="c4">pc</font>,<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"> ( 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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">p1</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c2">p2</font> or 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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">p1</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c2">p2</font> )</span><br/><div class="add"><a NAME="E2:57_1_2_1_2_5_2_1_2"/><b>then </b>
(  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c2">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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">p1</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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#K35" title="SIN_COS:func.35">PI</a> </span>)</span> or  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c2">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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">p1</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</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#K35" title="SIN_COS:func.35">PI</a> </span>)</span> )
 <b>by </b><i><a class="txt" href="euclid_6.html#E8:57_1_2_1_2"><i><font color="Green" title="E69">A23</font></i></a>, <a class="txt" href="euclid_6.html#E1:57_1_2_1_2_5_2"><i><font color="Green" title="E70">A25</font></i></a></i>;<br/><b>hence </b><a NAME="E3:57_1_2_1_2_5_2_1_2"/>
( 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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">p1</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c2">p2</font> or 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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">p1</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c2">p2</font> )
 <b>by </b><i><a class="txt" href="euclid_6.html#E6"><i><font color="Green" title="E6">Lm3</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="57_1_2_1_2_5_2_1_3"><b>suppose </b></a><a NAME="E1:57_1_2_1_2_5_2_1_3"/>
( 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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">p1</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c8">p3</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c8">p3</font> &amp; 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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="c8">p3</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <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="c8">p3</font>,<font color="Maroon" title="c4">pc</font>,<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"> ( 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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">p1</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c2">p2</font> or 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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">p1</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c2">p2</font> )</span><br/><div class="add"><a NAME="E2:57_1_2_1_2_5_2_1_3"/><b>then </b>
 <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c2">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <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="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c8">p3</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c8">p3</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default">6 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span>
 <b>by </b><i><a class="txt" href="euclid_6.html#E1:57_1_2_1_2_5_2"><i><font color="Green" title="E70">A25</font></i></a></i>;<br/><b>hence </b><a NAME="E3:57_1_2_1_2_5_2_1_3"/>
( 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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">p1</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c2">p2</font> or 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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">p1</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c2">p2</font> )
 <b>by </b><i><a class="txt" href="euclid_6.html#E8:57_1_2_1_2"><i><font color="Green" title="E69">A23</font></i></a>, <a class="txt" href="euclid_6.html#E9"><i><font color="Green" title="E9">Lm6</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="57_1_2_1_2_5_2_1_4"><b>suppose </b></a><a NAME="E1:57_1_2_1_2_5_2_1_4"/>
( 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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">p1</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c8">p3</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c8">p3</font> &amp; 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c8">p3</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span> <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="c8">p3</font>,<font color="Maroon" title="c4">pc</font>,<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"> ( 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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">p1</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c2">p2</font> or 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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">p1</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c2">p2</font> )</span><br/><div class="add"><a NAME="E2:57_1_2_1_2_5_2_1_4"/><b>then </b>
 <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c2">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <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="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c8">p3</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c8">p3</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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#K35" title="SIN_COS:func.35">PI</a> </span>)</span>
 <b>by </b><i><a class="txt" href="euclid_6.html#E1:57_1_2_1_2_5_2"><i><font color="Green" title="E70">A25</font></i></a></i>;<br/><b>hence </b><a NAME="E3:57_1_2_1_2_5_2_1_4"/>
( 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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">p1</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c2">p2</font> or 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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">p1</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">pc</font>,<font color="Maroon" title="c2">p2</font> )
 <b>by </b><i><a class="txt" href="euclid_6.html#E8:57_1_2_1_2"><i><font color="Green" title="E69">A23</font></i></a>, <a class="txt" href="euclid_6.html#E8"><i><font color="Green" title="E8">Lm5</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><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

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