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<div><span class="kw">theorem </span><span class="lab"><font color="Green" title="E10">Th12</font></span>: <a NAME="T12"><span class="comment"><font color="firebrick">:: NUMBER11:12</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">m</font>, <font color="Olive" title="b2">n</font>, <font color="Olive" title="b3">b</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b3">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 holds <br/>( (  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="numeral1.html#K2" title="NUMERAL1:func.2">digits</a> (<font color="Olive" title="b1">m</font>,<font color="Olive" title="b3">b</font>)</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="numeral1.html#K2" title="NUMERAL1:func.2">digits</a> (<font color="Olive" title="b2">n</font>,<font color="Olive" title="b3">b</font>)</span>)</span> or (  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="numeral1.html#K2" title="NUMERAL1:func.2">digits</a> (<font color="Olive" title="b1">m</font>,<font color="Olive" title="b3">b</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="numeral1.html#K2" title="NUMERAL1:func.2">digits</a> (<font color="Olive" title="b2">n</font>,<font color="Olive" title="b3">b</font>)</span>)</span> &amp;  ex <font color="Olive" title="b4">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( <font color="Olive" title="b4">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="numeral1.html#K2" title="NUMERAL1:func.2">digits</a> (<font color="Olive" title="b1">m</font>,<font color="Olive" title="b3">b</font>)</span>)</span> &amp; <span class="p1">(<span class="default"><a href="numeral1.html#K2" title="NUMERAL1:func.2">digits</a> (<font color="Olive" title="b1">m</font>,<font color="Olive" title="b3">b</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="numeral1.html#K2" title="NUMERAL1:func.2">digits</a> (<font color="Olive" title="b2">n</font>,<font color="Olive" title="b3">b</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">i</font> &amp; (  for <font color="Olive" title="b5">j</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b5">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="numeral1.html#K2" title="NUMERAL1:func.2">digits</a> (<font color="Olive" title="b1">m</font>,<font color="Olive" title="b3">b</font>)</span>)</span> &amp; <span class="p1">(<span class="default"><a href="numeral1.html#K2" title="NUMERAL1:func.2">digits</a> (<font color="Olive" title="b1">m</font>,<font color="Olive" title="b3">b</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="numeral1.html#K2" title="NUMERAL1:func.2">digits</a> (<font color="Olive" title="b2">n</font>,<font color="Olive" title="b3">b</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">j</font> holds <br/><font color="Olive" title="b4">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b5">j</font> ) ) ) ) iff <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">n</font> )</div></div>
