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<div><span class="kw">theorem </span><span class="lab"><font color="Green" title="E23">Th18</font></span>: <a NAME="T18"><span class="comment"><font color="firebrick">:: NUMBER15:18</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">p</font> being   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a><br/>  for <font color="Olive" title="b2">m</font>, <font color="Olive" title="b3">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">p</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 4 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 holds <br/>( <span class="p1">(<span class="default"><a href="number15.html#K1" title="NUMBER15:func.1">divisors</a> (<span class="p2">(<span class="default"><font color="Olive" title="b1">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Olive" title="b3">n</font></span>)</span>,4,1)</span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="number15.html#K1" title="NUMBER15:func.1">divisors</a> (<span class="p2">(<span class="default"><font color="Olive" title="b1">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Olive" title="b3">n</font></span>)</span>,4,3)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b4">k</font> where <font color="Olive" title="b4">k</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : <font color="Olive" title="b4">k</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Olive" title="b1">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Olive" title="b3">n</font> </span> } </span>  &amp; ( <font color="Olive" title="b3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b2">m</font> implies (  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="number15.html#K1" title="NUMBER15:func.1">divisors</a> (<span class="p2">(<span class="default"><font color="Olive" title="b1">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Olive" title="b3">n</font></span>)</span>,4,1)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp;  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="number15.html#K1" title="NUMBER15:func.1">divisors</a> (<span class="p2">(<span class="default"><font color="Olive" title="b1">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Olive" title="b3">n</font></span>)</span>,4,3)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">m</font> ) ) &amp; ( <font color="Olive" title="b3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 implies (  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="number15.html#K1" title="NUMBER15:func.1">divisors</a> (<span class="p2">(<span class="default"><font color="Olive" title="b1">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Olive" title="b3">n</font></span>)</span>,4,1)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp;  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="number15.html#K1" title="NUMBER15:func.1">divisors</a> (<span class="p2">(<span class="default"><font color="Olive" title="b1">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Olive" title="b3">n</font></span>)</span>,4,3)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 ) ) )</div></div>
