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<div><span class="kw">theorem </span><span class="lab"><font color="Green" title="E15">Th6</font></span>: <a NAME="T6"><span class="comment"><font color="firebrick">:: METRIC_1:6</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">M</font> being    <a href="metric_1.html#L1" title="METRIC_1:struct.1">MetrStruct</a>   st (  for <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</font>, <font color="Olive" title="b4">c</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b1">M</font> holds <br/> ( (  <a href="metric_1.html#K1" title="METRIC_1:func.1">dist</a> (<font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  implies <font color="Olive" title="b2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">b</font> ) &amp; ( <font color="Olive" title="b2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">b</font> implies  <a href="metric_1.html#K1" title="METRIC_1:func.1">dist</a> (<font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  ) &amp;  <a href="metric_1.html#K1" title="METRIC_1:func.1">dist</a> (<font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K1" title="METRIC_1:func.1">dist</a> (<font color="Olive" title="b3">b</font>,<font color="Olive" title="b2">a</font>) &amp;  <a href="metric_1.html#K1" title="METRIC_1:func.1">dist</a> (<font color="Olive" title="b2">a</font>,<font color="Olive" title="b4">c</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K1" title="METRIC_1:func.1">dist</a> (<font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">b</font>)</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K1" title="METRIC_1:func.1">dist</a> (<font color="Olive" title="b3">b</font>,<font color="Olive" title="b4">c</font>)</span>)</span> ) ) holds <br/><font color="Olive" title="b1">M</font> is   <a href="metric_1.html#NM1" title="METRIC_1:NM.1">MetrSpace</a></div></div>
