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<div><span class="kw">theorem </span><span class="lab"><font color="Green" title="E144">Th158</font></span>: <a NAME="T158"><span class="comment"><font color="firebrick">:: GLCOLO00:158</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">G2</font> being   <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a><br/>  for <font color="Olive" title="b2">v</font>, <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <br/>  for <font color="Olive" title="b4">w</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Olive" title="b1">G2</font><br/>  for <font color="Olive" title="b5">G1</font> being    <a href="glib_006.html#M4" title="GLIB_006:mode.4">addAdjVertex</a> of <font color="Olive" title="b1">G2</font>,<font color="Olive" title="b2">v</font>,<font color="Olive" title="b3">e</font>,<font color="Olive" title="b4">w</font><br/>  for <font color="Olive" title="b6">t1</font> being    <a href="glcolo00.html#M1" title="GLCOLO00:mode.1">TColoring</a> of <font color="Olive" title="b5">G1</font><br/>  for <font color="Olive" title="b7">t2</font> being    <a href="glcolo00.html#M1" title="GLCOLO00:mode.1">TColoring</a> of <font color="Olive" title="b1">G2</font><br/>  for <font color="Olive" title="b8">x</font>, <font color="Olive" title="b9">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st  not <font color="Olive" title="b3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Olive" title="b1">G2</font> &amp;  not <font color="Olive" title="b2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Olive" title="b1">G2</font> &amp; <font color="Olive" title="b6">t1</font> <a href="glcolo00.html#K7" title="GLCOLO00:func.7">_V</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b7">t2</font> <a href="glcolo00.html#K7" title="GLCOLO00:func.7">_V</a></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">v</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> <font color="Olive" title="b8">x</font></span>)</span> &amp; <font color="Olive" title="b6">t1</font> <a href="glcolo00.html#K8" title="GLCOLO00:func.8">_E</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b7">t2</font> <a href="glcolo00.html#K8" title="GLCOLO00:func.8">_E</a></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">e</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> <font color="Olive" title="b9">y</font></span>)</span> &amp;  not <font color="Olive" title="b9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p2">(<span class="default"><font color="Olive" title="b7">t2</font> <a href="glcolo00.html#K7" title="GLCOLO00:func.7">_V</a></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p2">(<span class="default"><font color="Olive" title="b7">t2</font> <a href="glcolo00.html#K8" title="GLCOLO00:func.8">_E</a></span>)</span></span>)</span> &amp; <font color="Olive" title="b8">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b9">y</font> &amp; <font color="Olive" title="b8">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Olive" title="b7">t2</font> <a href="glcolo00.html#K7" title="GLCOLO00:func.7">_V</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">w</font> &amp; <font color="Olive" title="b7">t2</font> is  <a href="glcolo00.html#V7" title="GLCOLO00:attr.7">proper</a>  holds <br/><font color="Olive" title="b6">t1</font> is  <a href="glcolo00.html#V7" title="GLCOLO00:attr.7">proper</a> </div></div>
