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<div><span class="kw">theorem </span><span class="lab"><font color="Green" title="E11">Th13</font></span>: <a NAME="T13"><span class="comment"><font color="firebrick">:: ZFREFLE1:13</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">W</font> being   <a href="classes2.html#NM1" title="CLASSES2:NM.1">Universe</a><br/>  for <font color="Olive" title="b2">D</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <br/>  for <font color="Olive" title="b3">Phi</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Olive" title="b2">D</font>,<span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p2">(<span class="default"><a href="ordinal1.html#K2" title="ORDINAL1:func.2">On</a> <font color="Olive" title="b1">W</font></span>)</span>,<span class="p2">(<span class="default"><a href="ordinal1.html#K2" title="ORDINAL1:func.2">On</a> <font color="Olive" title="b1">W</font></span>)</span>)</span>)</span>  st  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b2">D</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">W</font> holds <br/> ex <font color="Olive" title="b4">phi</font> being   <a href="ordinal4.html#NM2" title="ORDINAL4:NM.2">Ordinal-Sequence</a> of <font color="Olive" title="b1">W</font> st <br/>( <font color="Olive" title="b4">phi</font> is  <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a>  &amp; <font color="Olive" title="b4">phi</font> is  <a href="ordinal2.html#V3" title="ORDINAL2:attr.3">continuous</a>  &amp; <font color="Olive" title="b4">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Olive" title="b1">W</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Olive" title="b1">W</font> &amp; (  for <font color="Olive" title="b5">a</font> being   <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Olive" title="b1">W</font> holds  <font color="Olive" title="b4">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Olive" title="b5">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p1">(<span class="default"><span class="p2"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b4">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Olive" title="b5">a</font></span>)</span></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_5.html#K13" title="FUNCT_5:func.13">uncurry</a> <font color="Olive" title="b3">Phi</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <span class="p3"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Olive" title="b2">D</font>,<span class="p4"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><span class="p5">(<span class="default"><a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Olive" title="b5">a</font></span>)</span></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span></span>)</span> ) &amp; (  for <font color="Olive" title="b5">a</font> being   <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Olive" title="b1">W</font>  st <font color="Olive" title="b5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Olive" title="b1">W</font> &amp; <font color="Olive" title="b5">a</font> is  <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a>  holds <br/><font color="Olive" title="b4">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Olive" title="b5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">phi</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <font color="Olive" title="b5">a</font></span>)</span> ) )</div></div>
