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<div><span class="kw">theorem </span><span class="lab"><font color="Green" title="E50">Th31</font></span>: <a NAME="T31"><span class="comment"><font color="firebrick">:: EUCLID_6:31</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">p1</font>, <font color="Olive" title="b2">p2</font>, <font color="Olive" title="b3">p</font>, <font color="Olive" title="b4">pc</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><br/>  for <font color="Olive" title="b5">a</font>, <font color="Olive" title="b6">b</font>, <font color="Olive" title="b7">r</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Olive" title="b1">p1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jgraph_6.html#K5" title="JGRAPH_6:func.5">circle</a> (<font color="Olive" title="b5">a</font>,<font color="Olive" title="b6">b</font>,<font color="Olive" title="b7">r</font>) &amp; <font color="Olive" title="b2">p2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jgraph_6.html#K5" title="JGRAPH_6:func.5">circle</a> (<font color="Olive" title="b5">a</font>,<font color="Olive" title="b6">b</font>,<font color="Olive" title="b7">r</font>) &amp; <font color="Olive" title="b3">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jgraph_6.html#K5" title="JGRAPH_6:func.5">circle</a> (<font color="Olive" title="b5">a</font>,<font color="Olive" title="b6">b</font>,<font color="Olive" title="b7">r</font>) &amp; <font color="Olive" title="b4">pc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Olive" title="b5">a</font>,<font color="Olive" title="b6">b</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> &amp; <font color="Olive" title="b4">pc</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Olive" title="b3">p</font>,<font color="Olive" title="b2">p2</font>) &amp; <font color="Olive" title="b1">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">p</font> &amp;  not 2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Olive" title="b1">p1</font>,<font color="Olive" title="b3">p</font>,<font color="Olive" title="b2">p2</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Olive" title="b1">p1</font>,<font color="Olive" title="b4">pc</font>,<font color="Olive" title="b2">p2</font>) holds <br/>2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Olive" title="b1">p1</font>,<font color="Olive" title="b3">p</font>,<font color="Olive" title="b2">p2</font>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="sin_cos.html#K28" title="SIN_COS:func.28">PI</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Olive" title="b1">p1</font>,<font color="Olive" title="b4">pc</font>,<font color="Olive" title="b2">p2</font>)</div></div>
