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<div>:: <span class="kw">deftheorem </span><span class="lab"><font color="Green" title="E25">Def3</font></span>   defines <a href="nat_6.html#K1" title="NAT_6:func.1">LegendreSymbol</a> <a onclick="hs(this)" href="javascript:()">NAT_6:def 3 : <br/></a><span> for <font color="Olive" title="b1">p</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b2">a</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> holds <br/> ( ( <font color="Olive" title="b2">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Olive" title="b2">a</font> <a href="int_5.html#R1" title="INT_5:pred.1">is_quadratic_residue_mod</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 implies  <a href="nat_6.html#K1" title="NAT_6:func.1">LegendreSymbol</a> (<font color="Olive" title="b2">a</font>,<font color="Olive" title="b1">p</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) &amp; ( <font color="Olive" title="b1">p</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Olive" title="b2">a</font> implies  <a href="nat_6.html#K1" title="NAT_6:func.1">LegendreSymbol</a> (<font color="Olive" title="b2">a</font>,<font color="Olive" title="b1">p</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  ) &amp; ( <font color="Olive" title="b2">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Olive" title="b2">a</font> <a href="nat_6.html#NR1" title="NAT_6:NR.1">is_quadratic_non_residue_mod</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 implies  <a href="nat_6.html#K1" title="NAT_6:func.1">LegendreSymbol</a> (<font color="Olive" title="b2">a</font>,<font color="Olive" title="b1">p</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 ) );<br/></span></div>
