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<b>let </b><font color="Maroon" title="c1">p1</font>, <font color="Maroon" title="c2">p2</font>, <font color="Maroon" title="c3">p3</font>, <font color="Maroon" title="c4">p4</font>, <font color="Maroon" title="c5">p</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">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="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">p4</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="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c3">p3</font> &amp; <font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c4">p4</font> &amp; <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p4</font> <a href="zfmisc_1.html#R2" title="ZFMISC_1:pred.2">are_mutually_different</a>  holds <br/> <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">p4</font>,<font color="Maroon" title="c2">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c2">p2</font></span><br/><b>let </b><font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c7">b</font>, <font color="Maroon" title="c8">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="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">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="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">r</font> &amp; <font color="Maroon" title="c3">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="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">r</font> &amp; <font color="Maroon" title="c4">p4</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="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">r</font> &amp; <font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c3">p3</font> &amp; <font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c4">p4</font> &amp; <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p4</font> <a href="zfmisc_1.html#R2" title="ZFMISC_1:pred.2">are_mutually_different</a>  implies  <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">p4</font>,<font color="Maroon" title="c2">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c2">p2</font> )</span><br/>















<b>assume </b><a NAME="E1:62"/><i><font color="Green" title="E59">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="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">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="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">r</font> &amp; <font color="Maroon" title="c3">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="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">r</font> &amp; <font color="Maroon" title="c4">p4</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="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">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="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c3">p3</font> or  not <font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c4">p4</font> or  not <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p4</font> <a href="zfmisc_1.html#R2" title="ZFMISC_1:pred.2">are_mutually_different</a>  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">p4</font>,<font color="Maroon" title="c2">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c2">p2</font> )</span><br/>

<b>assume </b><a NAME="E2:62"/><i><font color="Green" title="E60">A2</font></i>: 
( <font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c3">p3</font> &amp; <font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c4">p4</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p4</font> <a href="zfmisc_1.html#R2" title="ZFMISC_1:pred.2">are_mutually_different</a>  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">p4</font>,<font color="Maroon" title="c2">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c2">p2</font> )</span><br/>

<b>assume </b><a NAME="E3:62"/>
<font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p4</font> <a href="zfmisc_1.html#R2" title="ZFMISC_1:pred.2">are_mutually_different</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">p4</font>,<font color="Maroon" title="c2">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c2">p2</font></span><br/>

<a NAME="E4:62"/><b>then </b><i><font color="Green" title="E61">A3</font></i>: 
( <font color="Maroon" title="c1">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">p2</font> &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">p3</font> &amp; <font color="Maroon" title="c1">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">p4</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">p3</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="c4">p4</font> &amp; <font color="Maroon" title="c3">p3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">p4</font> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#D6" title="ZFMISC_1:def.6">ZFMISC_1:def 6</a></i>;<br/>
<a NAME="E5:62"/><i><font color="Green" title="E62">A4</font></i>: 
 <a href="jgraph_6.html#K6" title="JGRAPH_6:func.6">inside_of_circle</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">r</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="jgraph_6.html#K5" title="JGRAPH_6:func.5">circle</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">r</font>
 
<b>by </b><i><a class="ref" href="topreal9.html#T54" title="TOPREAL9:th.54">TOPREAL9:54</a></i>;<br/>
<a NAME="E6:62"/><i><font color="Green" title="E63">A5</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c4">p4</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c4">p4</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="jgraph_6.html#K6" title="JGRAPH_6:func.6">inside_of_circle</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">r</font>
 
<b>by </b><i><a class="txt" href="#E1:62"><i><font color="Green" title="E59">A1</font></i></a>, <a class="ref" href="topreal9.html#T60" title="TOPREAL9:th.60">TOPREAL9:60</a></i>;<br/>
<a NAME="E7:62"/><i><font color="Green" title="E64">A6</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c3">p3</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c3">p3</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="jgraph_6.html#K6" title="JGRAPH_6:func.6">inside_of_circle</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">r</font>
 
<b>by </b><i><a class="txt" href="#E1:62"><i><font color="Green" title="E59">A1</font></i></a>, <a class="ref" href="topreal9.html#T60" title="TOPREAL9:th.60">TOPREAL9:60</a></i>;<br/>
<a NAME="E8:62"/><i><font color="Green" title="E65">A7</font></i>: 
( <font color="Maroon" title="c5">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">p1</font> &amp; <font color="Maroon" title="c5">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">p4</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:62_1"/><i><font color="Green" title="E66">A8</font></i>: 
( <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p1</font> or <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p4</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:62_1_1"/>
( <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p1</font> or <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p4</font> )
 </span><b>by </b><i><a class="txt" href="#E1:62_1"><i><font color="Green" title="E66">A8</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_1_1"><b>suppose </b></a><a NAME="E1:62_1_1_1"/><i><font color="Green" title="E67">A9</font></i>: 
<font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:62_1_1_1"/>
not <font color="Maroon" title="c1">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c4">p4</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <b>by </b><i><a class="txt" href="#E4:62"><i><font color="Green" title="E61">A3</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><a NAME="E3:62_1_1_1"/><b>then </b>
<font color="Maroon" title="c1">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c4">p4</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c4">p4</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <b>by </b><i><a class="txt" href="#E2:62"><i><font color="Green" title="E60">A2</font></i></a>, <a class="txt" href="#E1:62_1_1_1"><i><font color="Green" title="E67">A9</font></i></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>;<br/><a NAME="E4:62_1_1_1"/><b>then </b>
<font color="Maroon" title="c1">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#K6" title="JGRAPH_6:func.6">inside_of_circle</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">r</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#K5" title="JGRAPH_6:func.5">circle</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">r</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:62"><i><font color="Green" title="E59">A1</font></i></a>, <a class="txt" href="#E6:62"><i><font color="Green" title="E63">A5</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E5:62_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E5:62"><i><font color="Green" title="E62">A4</font></i></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</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="62_1_1_2"><b>suppose </b></a><a NAME="E1:62_1_1_2"/><i><font color="Green" title="E67">A10</font></i>: 
<font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p4</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:62_1_1_2"/>
not <font color="Maroon" title="c4">p4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c3">p3</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <b>by </b><i><a class="txt" href="#E4:62"><i><font color="Green" title="E61">A3</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><a NAME="E3:62_1_1_2"/><b>then </b>
<font color="Maroon" title="c4">p4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c3">p3</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c3">p3</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <b>by </b><i><a class="txt" href="#E2:62"><i><font color="Green" title="E60">A2</font></i></a>, <a class="txt" href="#E1:62_1_1_2"><i><font color="Green" title="E67">A10</font></i></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>;<br/><a NAME="E4:62_1_1_2"/><b>then </b>
<font color="Maroon" title="c4">p4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#K6" title="JGRAPH_6:func.6">inside_of_circle</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">r</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#K5" title="JGRAPH_6:func.5">circle</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">r</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:62"><i><font color="Green" title="E59">A1</font></i></a>, <a class="txt" href="#E7:62"><i><font color="Green" title="E64">A6</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E5:62_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E5:62"><i><font color="Green" title="E62">A4</font></i></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</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>
<a NAME="E9:62"/><i><font color="Green" title="E66">A11</font></i>: 
( <font color="Maroon" title="c5">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">p2</font> &amp; <font color="Maroon" title="c5">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">p3</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:62_2"/><i><font color="Green" title="E67">A12</font></i>: 
( <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p2</font> or <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p3</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:62_2_1"/>
( <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p3</font> or <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p2</font> )
 </span><b>by </b><i><a class="txt" href="#E1:62_2"><i><font color="Green" title="E67">A12</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_2_1_1"><b>suppose </b></a><a NAME="E1:62_2_1_1"/><i><font color="Green" title="E68">A13</font></i>: 
<font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p3</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:62_2_1_1"/>
not <font color="Maroon" title="c3">p3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c4">p4</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <b>by </b><i><a class="txt" href="#E4:62"><i><font color="Green" title="E61">A3</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><a NAME="E3:62_2_1_1"/><b>then </b>
<font color="Maroon" title="c3">p3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c4">p4</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c4">p4</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <b>by </b><i><a class="txt" href="#E2:62"><i><font color="Green" title="E60">A2</font></i></a>, <a class="txt" href="#E1:62_2_1_1"><i><font color="Green" title="E68">A13</font></i></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>;<br/><a NAME="E4:62_2_1_1"/><b>then </b>
<font color="Maroon" title="c3">p3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#K6" title="JGRAPH_6:func.6">inside_of_circle</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">r</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#K5" title="JGRAPH_6:func.5">circle</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">r</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:62"><i><font color="Green" title="E59">A1</font></i></a>, <a class="txt" href="#E6:62"><i><font color="Green" title="E63">A5</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E5:62_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E5:62"><i><font color="Green" title="E62">A4</font></i></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</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="62_2_1_2"><b>suppose </b></a><a NAME="E1:62_2_1_2"/><i><font color="Green" title="E68">A14</font></i>: 
<font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:62_2_1_2"/>
not <font color="Maroon" title="c2">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c3">p3</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <b>by </b><i><a class="txt" href="#E4:62"><i><font color="Green" title="E61">A3</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><a NAME="E3:62_2_1_2"/><b>then </b>
<font color="Maroon" title="c2">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c3">p3</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c3">p3</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <b>by </b><i><a class="txt" href="#E2:62"><i><font color="Green" title="E60">A2</font></i></a>, <a class="txt" href="#E1:62_2_1_2"><i><font color="Green" title="E68">A14</font></i></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>;<br/><a NAME="E4:62_2_1_2"/><b>then </b>
<font color="Maroon" title="c2">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#K6" title="JGRAPH_6:func.6">inside_of_circle</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">r</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#K5" title="JGRAPH_6:func.5">circle</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">r</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:62"><i><font color="Green" title="E59">A1</font></i></a>, <a class="txt" href="#E7:62"><i><font color="Green" title="E64">A6</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E5:62_2_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E5:62"><i><font color="Green" title="E62">A4</font></i></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</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>
<a NAME="E10:62"/><i><font color="Green" title="E67">A15</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">p4</font>,<font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">p4</font>,<font color="Maroon" title="c2">p2</font>
 
<b>by </b><i><a class="txt" href="#E2:62"><i><font color="Green" title="E60">A2</font></i></a>, <a class="txt" href="#E8:62"><i><font color="Green" title="E65">A7</font></i></a>, <a class="ref" href="euclid_6.html#T10" target="_self" title="EUCLID_6:th.10">Th10</a></i>;<br/>
<a NAME="E11:62"/><i><font color="Green" title="E68">A16</font></i>: 
 <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c2">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c2">p2</font>
 
<b>by </b><i><a class="txt" href="#E2:62"><i><font color="Green" title="E60">A2</font></i></a>, <a class="txt" href="#E9:62"><i><font color="Green" title="E66">A11</font></i></a>, <a class="ref" href="euclid_6.html#T9" target="_self" title="EUCLID_6:th.9">Th9</a></i>;<br/>
<a NAME="E12:62"/><i><font color="Green" title="E69">A17</font></i>: 
 <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c4">p4</font>,<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c1">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c3">p3</font>
 
<b>by </b><i><a class="txt" href="#E2:62"><i><font color="Green" title="E60">A2</font></i></a>, <a class="txt" href="#E8:62"><i><font color="Green" title="E65">A7</font></i></a>, <a class="txt" href="#E9:62"><i><font color="Green" title="E66">A11</font></i></a>, <a class="ref" href="euclid_6.html#T15" target="_self" title="EUCLID_6:th.15">Th15</a></i>;<br/>
<a NAME="E13:62"/><i><font color="Green" title="E70">A18</font></i>: 
<font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">p4</font>,<font color="Maroon" title="c5">p</font> <a href="zfmisc_1.html#R1" title="ZFMISC_1:pred.1">are_mutually_different</a> 
 
<b>by </b><i><a class="txt" href="#E4:62"><i><font color="Green" title="E61">A3</font></i></a>, <a class="txt" href="#E8:62"><i><font color="Green" title="E65">A7</font></i></a>, <a class="ref" href="zfmisc_1.html#D5" title="ZFMISC_1:def.5">ZFMISC_1:def 5</a></i>;<br/>
<a NAME="E14:62"/><i><font color="Green" title="E71">A19</font></i>: 
<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c3">p3</font> <a href="zfmisc_1.html#R1" title="ZFMISC_1:pred.1">are_mutually_different</a> 
 
<b>by </b><i><a class="txt" href="#E4:62"><i><font color="Green" title="E61">A3</font></i></a>, <a class="txt" href="#E9:62"><i><font color="Green" title="E66">A11</font></i></a>, <a class="ref" href="zfmisc_1.html#D5" title="ZFMISC_1:def.5">ZFMISC_1:def 5</a></i>;<br/>
<a NAME="E15:62"/><i><font color="Green" title="E72">A20</font></i>: 
<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c2">p2</font> <a href="zfmisc_1.html#R1" title="ZFMISC_1:pred.1">are_mutually_different</a> 
 
<b>by </b><i><a class="txt" href="#E4:62"><i><font color="Green" title="E61">A3</font></i></a>, <a class="txt" href="#E9:62"><i><font color="Green" title="E66">A11</font></i></a>, <a class="ref" href="zfmisc_1.html#D5" title="ZFMISC_1:def.5">ZFMISC_1:def 5</a></i>;<br/>
<a NAME="E16:62"/><i><font color="Green" title="E73">A21</font></i>: 
<font color="Maroon" title="c4">p4</font>,<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c1">p1</font> <a href="zfmisc_1.html#R1" title="ZFMISC_1:pred.1">are_mutually_different</a> 
 
<b>by </b><i><a class="txt" href="#E4:62"><i><font color="Green" title="E61">A3</font></i></a>, <a class="txt" href="#E8:62"><i><font color="Green" title="E65">A7</font></i></a>, <a class="ref" href="zfmisc_1.html#D5" title="ZFMISC_1:def.5">ZFMISC_1:def 5</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:62_3"/>
(  <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">p4</font>,<font color="Maroon" title="c2">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c2">p2</font> or  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">p4</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">p3</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#K36" title="SIN_COS:func.36">PI</a>  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">p4</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">p3</font>,<font color="Maroon" title="c2">p2</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  )
 </span><b>by </b><i><a class="txt" href="#E1:62"><i><font color="Green" title="E59">A1</font></i></a>, <a class="txt" href="#E4:62"><i><font color="Green" title="E61">A3</font></i></a>, <a class="ref" href="euclid_6.html#T34" target="_self" title="EUCLID_6:th.34">Th34</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_3_1"><b>suppose </b></a><a NAME="E1:62_3_1"/>
 <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">p4</font>,<font color="Maroon" title="c2">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c3">p3</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">  <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">p4</font>,<font color="Maroon" title="c2">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c2">p2</font></span><br/><div class="add"><b>hence </b><a NAME="E2:62_3_1"/>
 <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">p4</font>,<font color="Maroon" title="c2">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c3">p3</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"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_3_2"><b>suppose </b></a><a NAME="E1:62_3_2"/><i><font color="Green" title="E74">A22</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">p4</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">p3</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#K36" title="SIN_COS:func.36">PI</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">p4</font>,<font color="Maroon" title="c2">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c2">p2</font></span><br/><div class="add"><a NAME="E2:62_3_2"/>
 <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">p3</font>,<font color="Maroon" title="c2">p2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a> 
 <b>by </b><i><a class="ref" href="complex2.html#T84" title="COMPLEX2:th.84">COMPLEX2:84</a></i>;<br/><a NAME="E3:62_3_2"/><b>then </b>
<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">p3</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#K36" title="SIN_COS:func.36">PI</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a> 
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E4:62_3_2"/><b>then </b>
 <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c3">p3</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a> 
 <b>by </b><i><a class="txt" href="#E10:62"><i><font color="Green" title="E67">A15</font></i></a>, <a class="txt" href="#E12:62"><i><font color="Green" title="E69">A17</font></i></a>, <a class="txt" href="#E13:62"><i><font color="Green" title="E70">A18</font></i></a>, <a class="txt" href="#E1:62_3_2"><i><font color="Green" title="E74">A22</font></i></a>, <a class="ref" href="euclid_6.html#T23" target="_self" title="EUCLID_6:th.23">Th23</a></i>;<br/><a NAME="E5:62_3_2"/><b>then </b><i><font color="Green" title="E75">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">p3</font>,<font color="Maroon" title="c2">p2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a> 
 <b>by </b><i><a class="txt" href="#E11:62"><i><font color="Green" title="E68">A16</font></i></a>, <a class="txt" href="#E14:62"><i><font color="Green" title="E71">A19</font></i></a>, <a class="ref" href="euclid_6.html#T23" target="_self" title="EUCLID_6:th.23">Th23</a></i>;<br/><a NAME="E6:62_3_2"/>
 <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">p4</font>,<font color="Maroon" title="c2">p2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="complex2.html#T84" title="COMPLEX2:th.84">COMPLEX2:84</a></i>;<br/><a NAME="E7:62_3_2"/><b>then </b>
<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">p3</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#K36" title="SIN_COS:func.36">PI</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a> 
 <b>by </b><i><a class="txt" href="#E1:62_3_2"><i><font color="Green" title="E74">A22</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E8:62_3_2"/><b>then </b><i><font color="Green" title="E76">A24</font></i>: 
<font color="Maroon" title="c3">p3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font>
 <b>by </b><i><a class="txt" href="#E5:62_3_2"><i><font color="Green" title="E75">A23</font></i></a>, <a class="ref" href="euclid_6.html#T11" target="_self" title="EUCLID_6:th.11">Th11</a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E9:62_3_2"/><i><font color="Green" title="E77">A25</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="jgraph_6.html#K6" title="JGRAPH_6:func.6">inside_of_circle</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">r</font>
 <b>by </b><i><a class="txt" href="#E1:62"><i><font color="Green" title="E59">A1</font></i></a>, <a class="ref" href="topreal9.html#T60" title="TOPREAL9:th.60">TOPREAL9:60</a></i>;<br/><a NAME="E10:62_3_2"/><i><font color="Green" title="E78">A26</font></i>: 
 <a href="jgraph_6.html#K6" title="JGRAPH_6:func.6">inside_of_circle</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">r</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="jgraph_6.html#K5" title="JGRAPH_6:func.5">circle</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">r</font>
 <b>by </b><i><a class="ref" href="topreal9.html#T54" title="TOPREAL9:th.54">TOPREAL9:54</a></i>;<br/><a NAME="E11:62_3_2"/>
not <font color="Maroon" title="c3">p3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <b>by </b><i><a class="txt" href="#E4:62"><i><font color="Green" title="E61">A3</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><a NAME="E12:62_3_2"/><b>then </b>
<font color="Maroon" title="c3">p3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <b>by </b><i><a class="txt" href="#E8:62_3_2"><i><font color="Green" title="E76">A24</font></i></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>;<br/><a NAME="E13:62_3_2"/><b>then </b>
<font color="Maroon" title="c3">p3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#K6" title="JGRAPH_6:func.6">inside_of_circle</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">r</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#K5" title="JGRAPH_6:func.5">circle</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">r</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:62"><i><font color="Green" title="E59">A1</font></i></a>, <a class="txt" href="#E9:62_3_2"><i><font color="Green" title="E77">A25</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E14:62_3_2"/>
 <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">p4</font>,<font color="Maroon" title="c2">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c2">p2</font>
 <b>by </b><i><a class="txt" href="#E10:62_3_2"><i><font color="Green" title="E78">A26</font></i></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</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="62_3_3"><b>suppose </b></a><a NAME="E1:62_3_3"/><i><font color="Green" title="E74">A27</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">p4</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">p3</font>,<font color="Maroon" title="c2">p2</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">p4</font>,<font color="Maroon" title="c2">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c2">p2</font></span><br/><div class="add"><a NAME="E2:62_3_3"/>
 <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">p4</font>,<font color="Maroon" title="c2">p2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a> 
 <b>by </b><i><a class="ref" href="complex2.html#T84" title="COMPLEX2:th.84">COMPLEX2:84</a></i>;<br/><a NAME="E3:62_3_3"/><b>then </b>
<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">p4</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#K36" title="SIN_COS:func.36">PI</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a> 
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E4:62_3_3"/><b>then </b>
 <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c4">p4</font>,<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c1">p1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a> 
 <b>by </b><i><a class="txt" href="#E11:62"><i><font color="Green" title="E68">A16</font></i></a>, <a class="txt" href="#E12:62"><i><font color="Green" title="E69">A17</font></i></a>, <a class="txt" href="#E15:62"><i><font color="Green" title="E72">A20</font></i></a>, <a class="txt" href="#E1:62_3_3"><i><font color="Green" title="E74">A27</font></i></a>, <a class="ref" href="euclid_6.html#T23" target="_self" title="EUCLID_6:th.23">Th23</a></i>;<br/><a NAME="E5:62_3_3"/><b>then </b><i><font color="Green" title="E75">A28</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">p4</font>,<font color="Maroon" title="c2">p2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a> 
 <b>by </b><i><a class="txt" href="#E10:62"><i><font color="Green" title="E67">A15</font></i></a>, <a class="txt" href="#E16:62"><i><font color="Green" title="E73">A21</font></i></a>, <a class="ref" href="euclid_6.html#T23" target="_self" title="EUCLID_6:th.23">Th23</a></i>;<br/><a NAME="E6:62_3_3"/>
 <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">p3</font>,<font color="Maroon" title="c2">p2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="complex2.html#T84" title="COMPLEX2:th.84">COMPLEX2:84</a></i>;<br/><a NAME="E7:62_3_3"/><b>then </b>
<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="c4">p4</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#K36" title="SIN_COS:func.36">PI</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a> 
 <b>by </b><i><a class="txt" href="#E1:62_3_3"><i><font color="Green" title="E74">A27</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E8:62_3_3"/><b>then </b><i><font color="Green" title="E76">A29</font></i>: 
<font color="Maroon" title="c4">p4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font>
 <b>by </b><i><a class="txt" href="#E5:62_3_3"><i><font color="Green" title="E75">A28</font></i></a>, <a class="ref" href="euclid_6.html#T11" target="_self" title="EUCLID_6:th.11">Th11</a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E9:62_3_3"/><i><font color="Green" title="E77">A30</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="jgraph_6.html#K6" title="JGRAPH_6:func.6">inside_of_circle</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">r</font>
 <b>by </b><i><a class="txt" href="#E1:62"><i><font color="Green" title="E59">A1</font></i></a>, <a class="ref" href="topreal9.html#T60" title="TOPREAL9:th.60">TOPREAL9:60</a></i>;<br/><a NAME="E10:62_3_3"/><i><font color="Green" title="E78">A31</font></i>: 
 <a href="jgraph_6.html#K6" title="JGRAPH_6:func.6">inside_of_circle</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">r</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="jgraph_6.html#K5" title="JGRAPH_6:func.5">circle</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">r</font>
 <b>by </b><i><a class="ref" href="topreal9.html#T54" title="TOPREAL9:th.54">TOPREAL9:54</a></i>;<br/><a NAME="E11:62_3_3"/>
not <font color="Maroon" title="c4">p4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <b>by </b><i><a class="txt" href="#E4:62"><i><font color="Green" title="E61">A3</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><a NAME="E12:62_3_3"/><b>then </b>
<font color="Maroon" title="c4">p4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <b>by </b><i><a class="txt" href="#E8:62_3_3"><i><font color="Green" title="E76">A29</font></i></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>;<br/><a NAME="E13:62_3_3"/><b>then </b>
<font color="Maroon" title="c4">p4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#K6" title="JGRAPH_6:func.6">inside_of_circle</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">r</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#K5" title="JGRAPH_6:func.5">circle</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">r</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:62"><i><font color="Green" title="E59">A1</font></i></a>, <a class="txt" href="#E9:62_3_3"><i><font color="Green" title="E77">A30</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E14:62_3_3"/>
 <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">p4</font>,<font color="Maroon" title="c2">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c2">p2</font>
 <b>by </b><i><a class="txt" href="#E10:62_3_3"><i><font color="Green" title="E78">A31</font></i></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</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>

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