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<span class="kw">let </span><font color="Maroon" title="c1">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">p</font> being   <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a><br/>  for <font color="Olive" title="b2">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="ec_pf_1.html#NK2" title="EC_PF_1:NK.2">GF</a> <font color="Olive" title="b1">p</font></span>)</span>  st <font color="Olive" title="b2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c1">n</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/> <a href="ec_pf_1.html#K2" title="EC_PF_1:func.2">Lege_p</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">a</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><span class="kw">let </span><font color="Maroon" title="c2">p</font> be   <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="ec_pf_1.html#NK2" title="EC_PF_1:NK.2">GF</a> <font color="Maroon" title="c2">p</font></span>)</span>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c1">n</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/> <a href="ec_pf_1.html#K2" title="EC_PF_1:func.2">Lege_p</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><span class="kw">let </span><font color="Maroon" title="c3">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="ec_pf_1.html#NK2" title="EC_PF_1:NK.2">GF</a> <font color="Maroon" title="c2">p</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c1">n</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies  <a href="ec_pf_1.html#K2" title="EC_PF_1:func.2">Lege_p</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/>





<span class="kw">assume </span><a NAME="E1:56"/><span class="lab"><font color="Green" title="E42">AS</font></span>: 
( <font color="Maroon" title="c3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c1">n</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="ec_pf_1.html#K2" title="EC_PF_1:func.2">Lege_p</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/>

<span class="lab"><font color="Green" title="E43">P0</font></span>: <font color="Maroon" title="c1">n</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 2</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T59" title="INT_1:th.59">INT_1:59</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 2
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56"><span class="lab"><font color="Green" title="E42">AS</font></span></a></span>
;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c4">n1</font> = <font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2 as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> ;<br/>
<a NAME="E4:56"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="binom.html#K2" title="BINOM:func.2">|^</a> 2 is  <a href="ec_pf_1.html#V2" title="EC_PF_1:attr.2">quadratic_residue</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56"><span class="lab"><font color="Green" title="E42">AS</font></span></a>, <a class="ref" href="ec_pf_1.html#T34" target="_self" title="EC_PF_1:th.34">QR1</a>, <a class="ref" href="ec_pf_1.html#T25" target="_self" title="EC_PF_1:th.25">EX6</a></span>;<br/>
<a NAME="E5:56"/><span class="kw">then </span>
<font color="Maroon" title="c3">a</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Maroon" title="c1">n</font> is  <a href="ec_pf_1.html#V2" title="EC_PF_1:attr.2">quadratic_residue</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:56"><span class="lab"><font color="Green" title="E43">P0</font></span></a>, <a class="ref" href="ec_pf_1.html#T31" target="_self" title="EC_PF_1:th.31">EX12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:56"/>
 <a href="ec_pf_1.html#K2" title="EC_PF_1:func.2">Lege_p</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ec_pf_1.html#D5" target="_self" title="EC_PF_1:def.5">QRDef3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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