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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>;<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> ;<br/>















<b>assume </b><a NAME="E1:64"/><i><font color="Green" title="E60">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> )
 ;<br/>

<b>assume </b><a NAME="E2:64"/><i><font color="Green" title="E61">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> )
 ;<br/>

<a NAME="E3:64"/><i><font color="Green" title="E62">A3</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="c5">p</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="c5">p</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c1">p1</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#E2"><i><font color="Green" title="E2">Lm2</font></i></a></i>;<br/>
<a NAME="E4:64"/><i><font color="Green" title="E63">A4</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="c5">p</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="c5">p</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
 
<b>by </b><i><a class="txt" href="euclid_6.html#E2"><i><font color="Green" title="E2">Lm2</font></i></a></i>;<br/>
<a NAME="E5:64"/><i><font color="Green" title="E64">A5</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="c3">p3</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">p</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="c5">p</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c3">p3</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#E2"><i><font color="Green" title="E2">Lm2</font></i></a></i>;<br/>
<a NAME="E6:64"/><i><font color="Green" title="E65">A6</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="c4">p4</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">p</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="c5">p</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c4">p4</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#E2"><i><font color="Green" title="E2">Lm2</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:64_1"/>
(  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 <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>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1"><b>suppose </b></a><a NAME="E1:64_1_1"/><i><font color="Green" title="E66">A7</font></i>: 
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> 
 ;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:64_1_1_1"/>
( <font color="Maroon" title="c1">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p2</font> or <font color="Maroon" title="c1">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p3</font> or <font color="Maroon" title="c1">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p4</font> or <font color="Maroon" title="c2">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p3</font> or <font color="Maroon" title="c2">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p4</font> or <font color="Maroon" title="c3">p3</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="euclid_6.html#E1:64_1_1"><i><font color="Green" title="E66">A7</font></i></a>, <a class="ref" href="zfmisc_1.html#D6" title="ZFMISC_1:def.6">ZFMISC_1:def 6</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_1_1"><b>suppose </b></a><a NAME="E1:64_1_1_1_1"/>
<font color="Maroon" title="c1">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:64_1_1_1_1"/>
<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="c5">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c3">p3</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="c2">p2</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c4">p4</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#E1:64"><i><font color="Green" title="E60">A1</font></i></a>, <a class="txt" href="euclid_6.html#E2:64"><i><font color="Green" title="E61">A2</font></i></a>, <a class="txt" href="euclid_6.html#E60"><i><font color="Green" title="E56">Lm22</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_1_2"><b>suppose </b></a><a NAME="E1:64_1_1_1_2"/>
<font color="Maroon" title="c1">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p3</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:64_1_1_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="c5">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c3">p3</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="c2">p2</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c4">p4</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#E1:64"><i><font color="Green" title="E60">A1</font></i></a>, <a class="txt" href="euclid_6.html#E2:64"><i><font color="Green" title="E61">A2</font></i></a>, <a class="txt" href="euclid_6.html#E61"><i><font color="Green" title="E57">Lm23</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_1_3"><b>suppose </b></a><a NAME="E1:64_1_1_1_3"/>
<font color="Maroon" title="c1">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p4</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:64_1_1_1_3"/>
<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="c5">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c3">p3</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="c2">p2</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c4">p4</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#E1:64"><i><font color="Green" title="E60">A1</font></i></a>, <a class="txt" href="euclid_6.html#E2:64"><i><font color="Green" title="E61">A2</font></i></a>, <a class="txt" href="euclid_6.html#E4:64"><i><font color="Green" title="E63">A4</font></i></a>, <a class="txt" href="euclid_6.html#E6:64"><i><font color="Green" title="E65">A6</font></i></a>, <a class="txt" href="euclid_6.html#E60"><i><font color="Green" title="E56">Lm22</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_1_4"><b>suppose </b></a><a NAME="E1:64_1_1_1_4"/>
<font color="Maroon" title="c2">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p3</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:64_1_1_1_4"/>
<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="c5">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c3">p3</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="c2">p2</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c4">p4</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#E1:64"><i><font color="Green" title="E60">A1</font></i></a>, <a class="txt" href="euclid_6.html#E2:64"><i><font color="Green" title="E61">A2</font></i></a>, <a class="txt" href="euclid_6.html#E3:64"><i><font color="Green" title="E62">A3</font></i></a>, <a class="txt" href="euclid_6.html#E5:64"><i><font color="Green" title="E64">A5</font></i></a>, <a class="txt" href="euclid_6.html#E60"><i><font color="Green" title="E56">Lm22</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_1_5"><b>suppose </b></a><a NAME="E1:64_1_1_1_5"/>
<font color="Maroon" title="c2">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p4</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:64_1_1_1_5"/>
<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="c5">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c3">p3</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="c2">p2</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c4">p4</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#E1:64"><i><font color="Green" title="E60">A1</font></i></a>, <a class="txt" href="euclid_6.html#E2:64"><i><font color="Green" title="E61">A2</font></i></a>, <a class="txt" href="euclid_6.html#E61"><i><font color="Green" title="E57">Lm23</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_1_6"><b>suppose </b></a><a NAME="E1:64_1_1_1_6"/>
<font color="Maroon" title="c3">p3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p4</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:64_1_1_1_6"/>
<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="c5">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c3">p3</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="c2">p2</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c4">p4</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#E1:64"><i><font color="Green" title="E60">A1</font></i></a>, <a class="txt" href="euclid_6.html#E2:64"><i><font color="Green" title="E61">A2</font></i></a>, <a class="txt" href="euclid_6.html#E3:64"><i><font color="Green" title="E62">A3</font></i></a>, <a class="txt" href="euclid_6.html#E4:64"><i><font color="Green" title="E63">A4</font></i></a>, <a class="txt" href="euclid_6.html#E6:64"><i><font color="Green" title="E65">A6</font></i></a>, <a class="txt" href="euclid_6.html#E60"><i><font color="Green" title="E56">Lm22</font></i></a></i>;<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="64_1_2"><b>suppose </b></a><a NAME="E1:64_1_2"/><i><font color="Green" title="E66">A8</font></i>: 
<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> 
 ;<br/><div class="add"><a NAME="E2:64_1_2"/><b>then </b><i><font color="Green" title="E67">A9</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="E3:64_1_2"/><i><font color="Green" title="E68">A10</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="E4:64_1_2"/><i><font color="Green" title="E69">A11</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="euclid_6.html#E1:64"><i><font color="Green" title="E60">A1</font></i></a>, <a class="ref" href="topreal9.html#T60" title="TOPREAL9:th.60">TOPREAL9:60</a></i>;<br/><a NAME="E5:64_1_2"/><i><font color="Green" title="E70">A12</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="euclid_6.html#E1:64"><i><font color="Green" title="E60">A1</font></i></a>, <a class="ref" href="topreal9.html#T60" title="TOPREAL9:th.60">TOPREAL9:60</a></i>;<br/><a NAME="E6:64_1_2"/><i><font color="Green" title="E71">A13</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="64_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:64_1_2_1"/><i><font color="Green" title="E72">A14</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> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:64_1_2_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="euclid_6.html#E1:64_1_2_1"><i><font color="Green" title="E72">A14</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:64_1_2_1_1_1"/><i><font color="Green" title="E73">A15</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>
 ;<br/><div class="add"><a NAME="E2:64_1_2_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="euclid_6.html#E2:64_1_2"><i><font color="Green" title="E67">A9</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><a NAME="E3:64_1_2_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="euclid_6.html#E2:64"><i><font color="Green" title="E61">A2</font></i></a>, <a class="txt" href="euclid_6.html#E1:64_1_2_1_1_1"><i><font color="Green" title="E73">A15</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E4:64_1_2_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="euclid_6.html#E1:64"><i><font color="Green" title="E60">A1</font></i></a>, <a class="txt" href="euclid_6.html#E4:64_1_2"><i><font color="Green" title="E69">A11</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E5:64_1_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="euclid_6.html#E3:64_1_2"><i><font color="Green" title="E68">A10</font></i></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:64_1_2_1_1_2"/><i><font color="Green" title="E73">A16</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>
 ;<br/><div class="add"><a NAME="E2:64_1_2_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="euclid_6.html#E2:64_1_2"><i><font color="Green" title="E67">A9</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><a NAME="E3:64_1_2_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="euclid_6.html#E2:64"><i><font color="Green" title="E61">A2</font></i></a>, <a class="txt" href="euclid_6.html#E1:64_1_2_1_1_2"><i><font color="Green" title="E73">A16</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E4:64_1_2_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="euclid_6.html#E1:64"><i><font color="Green" title="E60">A1</font></i></a>, <a class="txt" href="euclid_6.html#E5:64_1_2"><i><font color="Green" title="E70">A12</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E5:64_1_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="euclid_6.html#E3:64_1_2"><i><font color="Green" title="E68">A10</font></i></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E7:64_1_2"/><i><font color="Green" title="E72">A17</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="64_1_2_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:64_1_2_2"/><i><font color="Green" title="E73">A18</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> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:64_1_2_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="euclid_6.html#E1:64_1_2_2"><i><font color="Green" title="E73">A18</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:64_1_2_2_1_1"/><i><font color="Green" title="E74">A19</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>
 ;<br/><div class="add"><a NAME="E2:64_1_2_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="euclid_6.html#E2:64_1_2"><i><font color="Green" title="E67">A9</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><a NAME="E3:64_1_2_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="euclid_6.html#E2:64"><i><font color="Green" title="E61">A2</font></i></a>, <a class="txt" href="euclid_6.html#E1:64_1_2_2_1_1"><i><font color="Green" title="E74">A19</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E4:64_1_2_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="euclid_6.html#E1:64"><i><font color="Green" title="E60">A1</font></i></a>, <a class="txt" href="euclid_6.html#E4:64_1_2"><i><font color="Green" title="E69">A11</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E5:64_1_2_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="euclid_6.html#E3:64_1_2"><i><font color="Green" title="E68">A10</font></i></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:64_1_2_2_1_2"/><i><font color="Green" title="E74">A20</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>
 ;<br/><div class="add"><a NAME="E2:64_1_2_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="euclid_6.html#E2:64_1_2"><i><font color="Green" title="E67">A9</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><a NAME="E3:64_1_2_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="euclid_6.html#E2:64"><i><font color="Green" title="E61">A2</font></i></a>, <a class="txt" href="euclid_6.html#E1:64_1_2_2_1_2"><i><font color="Green" title="E74">A20</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E4:64_1_2_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="euclid_6.html#E1:64"><i><font color="Green" title="E60">A1</font></i></a>, <a class="txt" href="euclid_6.html#E5:64_1_2"><i><font color="Green" title="E70">A12</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E5:64_1_2_2_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="euclid_6.html#E3:64_1_2"><i><font color="Green" title="E68">A10</font></i></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E8:64_1_2"/><i><font color="Green" title="E73">A21</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="euclid_6.html#E2:64"><i><font color="Green" title="E61">A2</font></i></a>, <a class="txt" href="euclid_6.html#E6:64_1_2"><i><font color="Green" title="E71">A13</font></i></a>, <a class="ref" href="euclid_6.html#T10" target="_self" title="EUCLID_6:th.10">Th10</a></i>;<br/><a NAME="E9:64_1_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="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="euclid_6.html#E2:64"><i><font color="Green" title="E61">A2</font></i></a>, <a class="txt" href="euclid_6.html#E7:64_1_2"><i><font color="Green" title="E72">A17</font></i></a>, <a class="ref" href="euclid_6.html#T9" target="_self" title="EUCLID_6:th.9">Th9</a></i>;<br/><a NAME="E10:64_1_2"/><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="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="euclid_6.html#E2:64"><i><font color="Green" title="E61">A2</font></i></a>, <a class="txt" href="euclid_6.html#E6:64_1_2"><i><font color="Green" title="E71">A13</font></i></a>, <a class="txt" href="euclid_6.html#E7:64_1_2"><i><font color="Green" title="E72">A17</font></i></a>, <a class="ref" href="euclid_6.html#T15" target="_self" title="EUCLID_6:th.15">Th15</a></i>;<br/><a NAME="E11:64_1_2"/><i><font color="Green" title="E76">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">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="euclid_6.html#E1:64"><i><font color="Green" title="E60">A1</font></i></a>, <a class="txt" href="euclid_6.html#E2:64"><i><font color="Green" title="E61">A2</font></i></a>, <a class="txt" href="euclid_6.html#E1:64_1_2"><i><font color="Green" title="E66">A8</font></i></a>, <a class="ref" href="euclid_6.html#T36" target="_self" title="EUCLID_6:th.36">Th36</a></i>;<br/><a NAME="E12:64_1_2"/><i><font color="Green" title="E77">A25</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="euclid_6.html#E2:64_1_2"><i><font color="Green" title="E67">A9</font></i></a>, <a class="txt" href="euclid_6.html#E6:64_1_2"><i><font color="Green" title="E71">A13</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="E13:64_1_2"/><i><font color="Green" title="E78">A26</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#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_2_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:64_1_2_3"/>
 <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="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> 
 ;<br/>

<a NAME="E2:64_1_2_3"/><b>then </b>
<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="c4">p4</font>
 
<b>by </b><i><a class="ref" href="euclid_6.html#T11" target="_self" title="EUCLID_6:th.11">Th11</a></i>;<br/>
<b>hence </b><a NAME="E3:64_1_2_3"/>
contradiction
 <b>by </b><i><a class="txt" href="euclid_6.html#E1:64"><i><font color="Green" title="E60">A1</font></i></a>, <a class="txt" href="euclid_6.html#E2:64"><i><font color="Green" title="E61">A2</font></i></a>, <a class="txt" href="euclid_6.html#E2:64_1_2"><i><font color="Green" title="E67">A9</font></i></a>, <a class="txt" href="euclid_6.html#E6:64_1_2"><i><font color="Green" title="E71">A13</font></i></a>, <a class="ref" href="euclid_6.html#T30" target="_self" title="EUCLID_6:th.30">Th30</a></i>;<br/>


</div><b>end;</b></div><a NAME="E14:64_1_2"/><i><font color="Green" title="E79">A27</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="c1">p1</font>,<font color="Maroon" title="c4">p4</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_2_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:64_1_2_4"/>
 <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">p4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> 
 ;<br/>

<a NAME="E2:64_1_2_4"/><b>then </b>
 <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">p4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> 
 
<b>by </b><i><a class="txt" href="euclid_6.html#E2:64"><i><font color="Green" title="E61">A2</font></i></a>, <a class="txt" href="euclid_6.html#E6:64_1_2"><i><font color="Green" title="E71">A13</font></i></a>, <a class="ref" href="euclid_6.html#T9" target="_self" title="EUCLID_6:th.9">Th9</a></i>;<br/>
<b>hence </b><a NAME="E3:64_1_2_4"/>
contradiction
 <b>by </b><i><a class="txt" href="euclid_6.html#E1:64"><i><font color="Green" title="E60">A1</font></i></a>, <a class="txt" href="euclid_6.html#E2:64_1_2"><i><font color="Green" title="E67">A9</font></i></a>, <a class="ref" href="euclid_6.html#T35" target="_self" title="EUCLID_6:th.35">Th35</a></i>;<br/>


</div><b>end;</b></div><a NAME="E15:64_1_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="c5">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_2_5"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:64_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">p4</font>,<font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> 
 ;<br/>

<a NAME="E2:64_1_2_5"/><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">p4</font>,<font color="Maroon" title="c2">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> 
 
<b>by </b><i><a class="txt" href="euclid_6.html#E2:64"><i><font color="Green" title="E61">A2</font></i></a>, <a class="txt" href="euclid_6.html#E6:64_1_2"><i><font color="Green" title="E71">A13</font></i></a>, <a class="ref" href="euclid_6.html#T10" target="_self" title="EUCLID_6:th.10">Th10</a></i>;<br/>
<b>hence </b><a NAME="E3:64_1_2_5"/>
contradiction
 <b>by </b><i><a class="txt" href="euclid_6.html#E1:64"><i><font color="Green" title="E60">A1</font></i></a>, <a class="txt" href="euclid_6.html#E2:64_1_2"><i><font color="Green" title="E67">A9</font></i></a>, <a class="ref" href="euclid_6.html#T35" target="_self" title="EUCLID_6:th.35">Th35</a></i>;<br/>


</div><b>end;</b></div><a NAME="E16:64_1_2"/><b>then </b><i><font color="Green" title="E80">A28</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="euclid_6.html#NR2" title="EUCLID_6:NR.2">is_a_triangle</a> 
 <b>by </b><i><a class="txt" href="euclid_6.html#E12:64_1_2"><i><font color="Green" title="E77">A25</font></i></a>, <a class="txt" href="euclid_6.html#E13:64_1_2"><i><font color="Green" title="E78">A26</font></i></a>, <a class="txt" href="euclid_6.html#E14:64_1_2"><i><font color="Green" title="E79">A27</font></i></a>, <a class="ref" href="euclid_6.html#T20" target="_self" title="EUCLID_6:th.20">Th20</a></i>;<br/><a NAME="E17:64_1_2"/><i><font color="Green" title="E81">A29</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="euclid_6.html#E2:64_1_2"><i><font color="Green" title="E67">A9</font></i></a>, <a class="txt" href="euclid_6.html#E7:64_1_2"><i><font color="Green" title="E72">A17</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="E18:64_1_2"/><i><font color="Green" title="E82">A30</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="c5">p</font>,<font color="Maroon" title="c3">p3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> 
 <b>by </b><i><a class="txt" href="euclid_6.html#E2:64"><i><font color="Green" title="E61">A2</font></i></a>, <a class="txt" href="euclid_6.html#E6:64_1_2"><i><font color="Green" title="E71">A13</font></i></a>, <a class="txt" href="euclid_6.html#E7:64_1_2"><i><font color="Green" title="E72">A17</font></i></a>, <a class="txt" href="euclid_6.html#E13:64_1_2"><i><font color="Green" title="E78">A26</font></i></a>, <a class="ref" href="euclid_6.html#T15" target="_self" title="EUCLID_6:th.15">Th15</a></i>;<br/><a NAME="E19:64_1_2"/><i><font color="Green" title="E83">A31</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#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_2_6"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:64_1_2_6"/>
 <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="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> 
 ;<br/>

<a NAME="E2:64_1_2_6"/><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="c3">p3</font>,<font color="Maroon" title="c2">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> 
 
<b>by </b><i><a class="txt" href="euclid_6.html#E2:64"><i><font color="Green" title="E61">A2</font></i></a>, <a class="txt" href="euclid_6.html#E7:64_1_2"><i><font color="Green" title="E72">A17</font></i></a>, <a class="ref" href="euclid_6.html#T9" target="_self" title="EUCLID_6:th.9">Th9</a></i>;<br/>
<b>hence </b><a NAME="E3:64_1_2_6"/>
contradiction
 <b>by </b><i><a class="txt" href="euclid_6.html#E1:64"><i><font color="Green" title="E60">A1</font></i></a>, <a class="txt" href="euclid_6.html#E2:64_1_2"><i><font color="Green" title="E67">A9</font></i></a>, <a class="ref" href="euclid_6.html#T35" target="_self" title="EUCLID_6:th.35">Th35</a></i>;<br/>


</div><b>end;</b></div><a NAME="E20:64_1_2"/>
 <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c5">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_2_7"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:64_1_2_7"/>
 <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> 
 ;<br/>

<a NAME="E2:64_1_2_7"/><b>then </b>
 <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c4">p4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> 
 
<b>by </b><i><a class="txt" href="euclid_6.html#E2:64"><i><font color="Green" title="E61">A2</font></i></a>, <a class="txt" href="euclid_6.html#E7:64_1_2"><i><font color="Green" title="E72">A17</font></i></a>, <a class="ref" href="euclid_6.html#T10" target="_self" title="EUCLID_6:th.10">Th10</a></i>;<br/>
<b>hence </b><a NAME="E3:64_1_2_7"/>
contradiction
 <b>by </b><i><a class="txt" href="euclid_6.html#E1:64"><i><font color="Green" title="E60">A1</font></i></a>, <a class="txt" href="euclid_6.html#E2:64_1_2"><i><font color="Green" title="E67">A9</font></i></a>, <a class="ref" href="euclid_6.html#T35" target="_self" title="EUCLID_6:th.35">Th35</a></i>;<br/>


</div><b>end;</b></div><a NAME="E21:64_1_2"/><b>then </b>
<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c3">p3</font> <a href="euclid_6.html#NR2" title="EUCLID_6:NR.2">is_a_triangle</a> 
 <b>by </b><i><a class="txt" href="euclid_6.html#E17:64_1_2"><i><font color="Green" title="E81">A29</font></i></a>, <a class="txt" href="euclid_6.html#E18:64_1_2"><i><font color="Green" title="E82">A30</font></i></a>, <a class="txt" href="euclid_6.html#E19:64_1_2"><i><font color="Green" title="E83">A31</font></i></a>, <a class="ref" href="euclid_6.html#T20" target="_self" title="EUCLID_6:th.20">Th20</a></i>;<br/><b>hence </b><a NAME="E22:64_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="c5">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c3">p3</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="c2">p2</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c5">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c4">p4</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#E3:64"><i><font color="Green" title="E62">A3</font></i></a>, <a class="txt" href="euclid_6.html#E4:64"><i><font color="Green" title="E63">A4</font></i></a>, <a class="txt" href="euclid_6.html#E5:64"><i><font color="Green" title="E64">A5</font></i></a>, <a class="txt" href="euclid_6.html#E6:64"><i><font color="Green" title="E65">A6</font></i></a>, <a class="txt" href="euclid_6.html#E8:64_1_2"><i><font color="Green" title="E73">A21</font></i></a>, <a class="txt" href="euclid_6.html#E9:64_1_2"><i><font color="Green" title="E74">A22</font></i></a>, <a class="txt" href="euclid_6.html#E10:64_1_2"><i><font color="Green" title="E75">A23</font></i></a>, <a class="txt" href="euclid_6.html#E11:64_1_2"><i><font color="Green" title="E76">A24</font></i></a>, <a class="txt" href="euclid_6.html#E16:64_1_2"><i><font color="Green" title="E80">A28</font></i></a>, <a class="ref" href="euclid_6.html#T22" target="_self" title="EUCLID_6:th.22">Th22</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
