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<div><span class="kw">theorem </span><a NAME="T14"><span class="comment"><font color="firebrick">:: GTARSKI2:14</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font>, <font color="Olive" title="b4">e</font>, <font color="Olive" title="b5">f</font>, <font color="Olive" title="b6">g</font> being   <a href="gtarski1.html#NM1" title="GTARSKI1:NM.1">POINT</a> of <a href="gtarski2.html#K2" title="GTARSKI2:func.2">TarskiEuclid2Space</a>  st  <a href="gtarski2.html#K4" title="GTARSKI2:func.4">Tn2TR</a> <font color="Olive" title="b1">a</font>, <a href="gtarski2.html#K4" title="GTARSKI2:func.4">Tn2TR</a> <font color="Olive" title="b2">b</font>, <a href="gtarski2.html#K4" title="GTARSKI2:func.4">Tn2TR</a> <font color="Olive" title="b3">c</font> <a href="euclid_6.html#NR1" title="EUCLID_6:NR.1">is_a_triangle</a>  &amp;  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<span class="p1">(<span class="default"><a href="gtarski2.html#K4" title="GTARSKI2:func.4">Tn2TR</a> <font color="Olive" title="b1">a</font></span>)</span>,<span class="p1">(<span class="default"><a href="gtarski2.html#K4" title="GTARSKI2:func.4">Tn2TR</a> <font color="Olive" title="b2">b</font></span>)</span>,<span class="p1">(<span class="default"><a href="gtarski2.html#K4" title="GTARSKI2:func.4">Tn2TR</a> <font color="Olive" title="b3">c</font></span>)</span>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="sin_cos.html#K28" title="SIN_COS:func.28">PI</a>  &amp;  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<span class="p1">(<span class="default"><a href="gtarski2.html#K4" title="GTARSKI2:func.4">Tn2TR</a> <font color="Olive" title="b4">e</font></span>)</span>,<span class="p1">(<span class="default"><a href="gtarski2.html#K4" title="GTARSKI2:func.4">Tn2TR</a> <font color="Olive" title="b3">c</font></span>)</span>,<span class="p1">(<span class="default"><a href="gtarski2.html#K4" title="GTARSKI2:func.4">Tn2TR</a> <font color="Olive" title="b1">a</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<span class="p2">(<span class="default"><a href="gtarski2.html#K4" title="GTARSKI2:func.4">Tn2TR</a> <font color="Olive" title="b2">b</font></span>)</span>,<span class="p2">(<span class="default"><a href="gtarski2.html#K4" title="GTARSKI2:func.4">Tn2TR</a> <font color="Olive" title="b3">c</font></span>)</span>,<span class="p2">(<span class="default"><a href="gtarski2.html#K4" title="GTARSKI2:func.4">Tn2TR</a> <font color="Olive" title="b1">a</font></span>)</span>)</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3 &amp;  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<span class="p1">(<span class="default"><a href="gtarski2.html#K4" title="GTARSKI2:func.4">Tn2TR</a> <font color="Olive" title="b3">c</font></span>)</span>,<span class="p1">(<span class="default"><a href="gtarski2.html#K4" title="GTARSKI2:func.4">Tn2TR</a> <font color="Olive" title="b1">a</font></span>)</span>,<span class="p1">(<span class="default"><a href="gtarski2.html#K4" title="GTARSKI2:func.4">Tn2TR</a> <font color="Olive" title="b4">e</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<span class="p2">(<span class="default"><a href="gtarski2.html#K4" title="GTARSKI2:func.4">Tn2TR</a> <font color="Olive" title="b3">c</font></span>)</span>,<span class="p2">(<span class="default"><a href="gtarski2.html#K4" title="GTARSKI2:func.4">Tn2TR</a> <font color="Olive" title="b1">a</font></span>)</span>,<span class="p2">(<span class="default"><a href="gtarski2.html#K4" title="GTARSKI2:func.4">Tn2TR</a> <font color="Olive" title="b2">b</font></span>)</span>)</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3 &amp;  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<span class="p1">(<span class="default"><a href="gtarski2.html#K4" title="GTARSKI2:func.4">Tn2TR</a> <font color="Olive" title="b1">a</font></span>)</span>,<span class="p1">(<span class="default"><a href="gtarski2.html#K4" title="GTARSKI2:func.4">Tn2TR</a> <font color="Olive" title="b2">b</font></span>)</span>,<span class="p1">(<span class="default"><a href="gtarski2.html#K4" title="GTARSKI2:func.4">Tn2TR</a> <font color="Olive" title="b5">f</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<span class="p2">(<span class="default"><a href="gtarski2.html#K4" title="GTARSKI2:func.4">Tn2TR</a> <font color="Olive" title="b1">a</font></span>)</span>,<span class="p2">(<span class="default"><a href="gtarski2.html#K4" title="GTARSKI2:func.4">Tn2TR</a> <font color="Olive" title="b2">b</font></span>)</span>,<span class="p2">(<span class="default"><a href="gtarski2.html#K4" title="GTARSKI2:func.4">Tn2TR</a> <font color="Olive" title="b3">c</font></span>)</span>)</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3 &amp;  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<span class="p1">(<span class="default"><a href="gtarski2.html#K4" title="GTARSKI2:func.4">Tn2TR</a> <font color="Olive" title="b5">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="gtarski2.html#K4" title="GTARSKI2:func.4">Tn2TR</a> <font color="Olive" title="b1">a</font></span>)</span>,<span class="p1">(<span class="default"><a href="gtarski2.html#K4" title="GTARSKI2:func.4">Tn2TR</a> <font color="Olive" title="b2">b</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<span class="p2">(<span class="default"><a href="gtarski2.html#K4" title="GTARSKI2:func.4">Tn2TR</a> <font color="Olive" title="b3">c</font></span>)</span>,<span class="p2">(<span class="default"><a href="gtarski2.html#K4" title="GTARSKI2:func.4">Tn2TR</a> <font color="Olive" title="b1">a</font></span>)</span>,<span class="p2">(<span class="default"><a href="gtarski2.html#K4" title="GTARSKI2:func.4">Tn2TR</a> <font color="Olive" title="b2">b</font></span>)</span>)</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3 &amp;  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<span class="p1">(<span class="default"><a href="gtarski2.html#K4" title="GTARSKI2:func.4">Tn2TR</a> <font color="Olive" title="b2">b</font></span>)</span>,<span class="p1">(<span class="default"><a href="gtarski2.html#K4" title="GTARSKI2:func.4">Tn2TR</a> <font color="Olive" title="b3">c</font></span>)</span>,<span class="p1">(<span class="default"><a href="gtarski2.html#K4" title="GTARSKI2:func.4">Tn2TR</a> <font color="Olive" title="b6">g</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<span class="p2">(<span class="default"><a href="gtarski2.html#K4" title="GTARSKI2:func.4">Tn2TR</a> <font color="Olive" title="b2">b</font></span>)</span>,<span class="p2">(<span class="default"><a href="gtarski2.html#K4" title="GTARSKI2:func.4">Tn2TR</a> <font color="Olive" title="b3">c</font></span>)</span>,<span class="p2">(<span class="default"><a href="gtarski2.html#K4" title="GTARSKI2:func.4">Tn2TR</a> <font color="Olive" title="b1">a</font></span>)</span>)</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3 &amp;  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<span class="p1">(<span class="default"><a href="gtarski2.html#K4" title="GTARSKI2:func.4">Tn2TR</a> <font color="Olive" title="b6">g</font></span>)</span>,<span class="p1">(<span class="default"><a href="gtarski2.html#K4" title="GTARSKI2:func.4">Tn2TR</a> <font color="Olive" title="b2">b</font></span>)</span>,<span class="p1">(<span class="default"><a href="gtarski2.html#K4" title="GTARSKI2:func.4">Tn2TR</a> <font color="Olive" title="b3">c</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<span class="p2">(<span class="default"><a href="gtarski2.html#K4" title="GTARSKI2:func.4">Tn2TR</a> <font color="Olive" title="b1">a</font></span>)</span>,<span class="p2">(<span class="default"><a href="gtarski2.html#K4" title="GTARSKI2:func.4">Tn2TR</a> <font color="Olive" title="b2">b</font></span>)</span>,<span class="p2">(<span class="default"><a href="gtarski2.html#K4" title="GTARSKI2:func.4">Tn2TR</a> <font color="Olive" title="b3">c</font></span>)</span>)</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 3 holds <br/>(  <a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<font color="Olive" title="b5">f</font>,<font color="Olive" title="b4">e</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<font color="Olive" title="b6">g</font>,<font color="Olive" title="b5">f</font>) &amp;  <a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<font color="Olive" title="b5">f</font>,<font color="Olive" title="b4">e</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<font color="Olive" title="b4">e</font>,<font color="Olive" title="b6">g</font>) &amp;  <a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<font color="Olive" title="b6">g</font>,<font color="Olive" title="b5">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<font color="Olive" title="b4">e</font>,<font color="Olive" title="b6">g</font>) )</div></div>
