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<div><span class="kw">theorem </span><span class="lab"><font color="Green" title="E28">Th25</font></span>: <a NAME="T25"><span class="comment"><font color="firebrick">:: NAT_6:25</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">a</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> holds <br/> ( (  <a href="nat_6.html#NK3" title="NAT_6:NK.3">Leg</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 implies ( <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="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> implies  <a href="nat_6.html#NK3" title="NAT_6:NK.3">Leg</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; (  <a href="nat_6.html#NK3" title="NAT_6:NK.3">Leg</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>  implies <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> ) &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#NK3" title="NAT_6:NK.3">Leg</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; (  <a href="nat_6.html#NK3" title="NAT_6:NK.3">Leg</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 implies ( <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="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> implies  <a href="nat_6.html#NK3" title="NAT_6:NK.3">Leg</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 ) )</div></div>
