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<div><span class="kw">theorem </span><span class="lab"><font color="Green" title="E42">Th54</font></span>: <a NAME="T54"><span class="comment"><font color="firebrick">:: CLASSES1:54</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">X</font>, <font color="Olive" title="b2">Y</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st (  for <font color="Olive" title="b3">Z</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">Z</font> &amp; <font color="Olive" title="b3">Z</font> is  <a href="ordinal1.html#V1" title="ORDINAL1:attr.1">epsilon-transitive</a>  holds <br/><font color="Olive" title="b2">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">Z</font> ) &amp; <font color="Olive" title="b1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">Y</font> &amp; <font color="Olive" title="b2">Y</font> is  <a href="ordinal1.html#V1" title="ORDINAL1:attr.1">epsilon-transitive</a>  holds <br/> <a href="classes1.html#K5" title="CLASSES1:func.5">the_transitive-closure_of</a> <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">Y</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="classes1.html#T53" target="_self" onmouseover="rs('classes1/T53')" onmouseout="rh()">Th53</a>, <a class="ref" href="classes1.html#T52" target="_self" onmouseover="rs('classes1/T52')" onmouseout="rh()">Th52</a></span>;<br/></div></div>
