<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">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>, <font color="Olive" title="b2">b</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="c1">p</font></span>)</span> holds   <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="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ec_pf_1.html#K2" title="EC_PF_1:func.2">Lege_p</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="ec_pf_1.html#K2" title="EC_PF_1:func.2">Lege_p</a> <font color="Olive" title="b2">b</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</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="c1">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">  <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="c2">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ec_pf_1.html#K2" title="EC_PF_1:func.2">Lege_p</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="ec_pf_1.html#K2" title="EC_PF_1:func.2">Lege_p</a> <font color="Maroon" title="c3">b</font></span>)</span></span><br/>





<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:55_1"/>
( <font color="Maroon" title="c2">a</font> is  <a href="ec_pf_1.html#V2" title="EC_PF_1:attr.2">quadratic_residue</a>  or  not <font color="Maroon" title="c2">a</font> is  <a href="ec_pf_1.html#V2" title="EC_PF_1:attr.2">quadratic_residue</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_1"><span class="kw">suppose </span></a><a NAME="E1:55_1_1"/><span class="lab"><font color="Green" title="E41">P0</font></span>: 
<font color="Maroon" title="c2">a</font> is  <a href="ec_pf_1.html#V2" title="EC_PF_1:attr.2">quadratic_residue</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="c2">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ec_pf_1.html#K2" title="EC_PF_1:func.2">Lege_p</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="ec_pf_1.html#K2" title="EC_PF_1:func.2">Lege_p</a> <font color="Maroon" title="c3">b</font></span>)</span></span><br/><div class="add"><a NAME="E2:55_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">P1</font></span>: 
 <a href="ec_pf_1.html#K2" title="EC_PF_1:func.2">Lege_p</a> <font color="Maroon" title="c2">a</font> <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#T36" target="_self" title="EC_PF_1:th.36">QR11</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:55_1_1_1"/>
( <font color="Maroon" title="c3">b</font> is  <a href="ec_pf_1.html#V2" title="EC_PF_1:attr.2">quadratic_residue</a>  or <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or ( <font color="Maroon" title="c3">b</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;  not <font color="Maroon" title="c3">b</font> is  <a href="ec_pf_1.html#V2" title="EC_PF_1:attr.2">quadratic_residue</a>  ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:55_1_1_1_1"/><span class="lab"><font color="Green" title="E43">A0</font></span>: 
<font color="Maroon" title="c3">b</font> is  <a href="ec_pf_1.html#V2" title="EC_PF_1:attr.2">quadratic_residue</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="c2">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ec_pf_1.html#K2" title="EC_PF_1:func.2">Lege_p</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="ec_pf_1.html#K2" title="EC_PF_1:func.2">Lege_p</a> <font color="Maroon" title="c3">b</font></span>)</span></span><br/><div class="add"><a NAME="E2:55_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A1</font></span>: 
 <a href="ec_pf_1.html#K2" title="EC_PF_1:func.2">Lege_p</a> <font color="Maroon" title="c3">b</font> <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#T36" target="_self" title="EC_PF_1:th.36">QR11</a></span>;<br/><a NAME="E3:55_1_1_1_1"/><span class="lab"><font color="Green" title="E45">A2</font></span>: 
( <font color="Maroon" title="c2">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;  ex <font color="Olive" title="b1">x</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="c1">p</font></span>)</span> st <font color="Olive" title="b1">x</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:55_1_1"><span class="lab"><font color="Green" title="E41">P0</font></span></a>, <a class="ref" href="ec_pf_1.html#D3" target="_self" title="EC_PF_1:def.3">QRDef1</a></span>;<br/><a NAME="E4:55_1_1_1_1"/>
( <font color="Maroon" title="c3">b</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;  ex <font color="Olive" title="b1">x</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="c1">p</font></span>)</span> st <font color="Olive" title="b1">x</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:55_1_1_1_1"><span class="lab"><font color="Green" title="E43">A0</font></span></a>, <a class="ref" href="ec_pf_1.html#D3" target="_self" title="EC_PF_1:def.3">QRDef1</a></span>;<br/><a NAME="E5:55_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A4</font></span>: 
<font color="Maroon" title="c2">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:55_1_1_1_1"><span class="lab"><font color="Green" title="E45">A2</font></span></a>, <a class="ref" href="ec_pf_1.html#T20" target="_self" title="EC_PF_1:th.20">GF8</a></span>;<br/><a NAME="E6:55_1_1_1_1"/>
 ex <font color="Olive" title="b1">x</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="c1">p</font></span>)</span> st <font color="Olive" title="b1">x</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c3">b</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">consider </span><font color="Maroon" title="c4">a1</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="c1">p</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E2:55_1_1_1_1_1"/><span class="lab"><font color="Green" title="E47">B0</font></span>: 
<font color="Maroon" title="c4">a1</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:55_1_1"><span class="lab"><font color="Green" title="E41">P0</font></span></a>, <a class="ref" href="ec_pf_1.html#D3" target="_self" title="EC_PF_1:def.3">QRDef1</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c5">b1</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="c1">p</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E4:55_1_1_1_1_1"/><span class="lab"><font color="Green" title="E48">B1</font></span>: 
<font color="Maroon" title="c5">b1</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:55_1_1_1_1"><span class="lab"><font color="Green" title="E43">A0</font></span></a>, <a class="ref" href="ec_pf_1.html#D3" target="_self" title="EC_PF_1:def.3">QRDef1</a></span>;<br/>
<a NAME="E5:55_1_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a1</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> 2</span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b1</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a1</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c5">b1</font></span>)</span> <a href="binom.html#K2" title="BINOM:func.2">|^</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ec_pf_1.html#T28" target="_self" title="EC_PF_1:th.28">EX9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:55_1_1_1_1_1"/>
 ex <font color="Olive" title="b1">x</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="c1">p</font></span>)</span> st <font color="Olive" title="b1">x</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c3">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:55_1_1_1_1_1"><span class="lab"><font color="Green" title="E47">B0</font></span></a>, <a class="txt" href="#E4:55_1_1_1_1_1"><span class="lab"><font color="Green" title="E48">B1</font></span></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/>


</div><span class="kw">end;</span></div><a NAME="E7:55_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c3">b</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="#E5:55_1_1_1_1"><span class="lab"><font color="Green" title="E46">A4</font></span></a>, <a class="ref" href="ec_pf_1.html#D3" target="_self" title="EC_PF_1:def.3">QRDef1</a></span>;<br/><span class="kw">hence </span><a NAME="E8:55_1_1_1_1"/>
 <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="c2">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ec_pf_1.html#K2" title="EC_PF_1:func.2">Lege_p</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="ec_pf_1.html#K2" title="EC_PF_1:func.2">Lege_p</a> <font color="Maroon" title="c3">b</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:55_1_1"><span class="lab"><font color="Green" title="E42">P1</font></span></a>, <a class="txt" href="#E2:55_1_1_1_1"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="ref" href="ec_pf_1.html#T36" target="_self" title="EC_PF_1:th.36">QR11</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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:55_1_1_1_2"/><span class="lab"><font color="Green" title="E43">A0</font></span>: 
<font color="Maroon" title="c3">b</font> <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="c2">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ec_pf_1.html#K2" title="EC_PF_1:func.2">Lege_p</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="ec_pf_1.html#K2" title="EC_PF_1:func.2">Lege_p</a> <font color="Maroon" title="c3">b</font></span>)</span></span><br/><div class="add"><a NAME="E2:55_1_1_1_2"/><span class="kw">then </span>
 <a href="ec_pf_1.html#K2" title="EC_PF_1:func.2">Lege_p</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <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>;<br/><span class="kw">hence </span><a NAME="E3:55_1_1_1_2"/>
 <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="c2">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ec_pf_1.html#K2" title="EC_PF_1:func.2">Lege_p</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="ec_pf_1.html#K2" title="EC_PF_1:func.2">Lege_p</a> <font color="Maroon" title="c3">b</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:55_1_1_1_2"><span class="lab"><font color="Green" title="E43">A0</font></span></a>, <a class="ref" href="ec_pf_1.html#T20" target="_self" title="EC_PF_1:th.20">GF8</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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:55_1_1_1_3"/><span class="lab"><font color="Green" title="E43">A0</font></span>: 
( <font color="Maroon" title="c3">b</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;  not <font color="Maroon" title="c3">b</font> is  <a href="ec_pf_1.html#V2" title="EC_PF_1:attr.2">quadratic_residue</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="c2">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ec_pf_1.html#K2" title="EC_PF_1:func.2">Lege_p</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="ec_pf_1.html#K2" title="EC_PF_1:func.2">Lege_p</a> <font color="Maroon" title="c3">b</font></span>)</span></span><br/><div class="add"><a NAME="E2:55_1_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A1</font></span>: 
 <a href="ec_pf_1.html#K2" title="EC_PF_1:func.2">Lege_p</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.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>;<br/><a NAME="E3:55_1_1_1_3"/><span class="lab"><font color="Green" title="E45">A2</font></span>: 
( <font color="Maroon" title="c2">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;  ex <font color="Olive" title="b1">x</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="c1">p</font></span>)</span> st <font color="Olive" title="b1">x</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:55_1_1"><span class="lab"><font color="Green" title="E41">P0</font></span></a>, <a class="ref" href="ec_pf_1.html#D3" target="_self" title="EC_PF_1:def.3">QRDef1</a></span>;<br/><a NAME="E4:55_1_1_1_3"/><span class="lab"><font color="Green" title="E46">A4</font></span>: 
<font color="Maroon" title="c2">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:55_1_1_1_3"><span class="lab"><font color="Green" title="E43">A0</font></span></a>, <a class="txt" href="#E3:55_1_1_1_3"><span class="lab"><font color="Green" title="E45">A2</font></span></a>, <a class="ref" href="ec_pf_1.html#T20" target="_self" title="EC_PF_1:th.20">GF8</a></span>;<br/><a NAME="E5:55_1_1_1_3"/>
 for <font color="Olive" title="b1">x</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="c1">p</font></span>)</span>  holds  not <font color="Olive" title="b1">x</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c3">b</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1_1_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">given </span><font color="Maroon" title="c4">xab</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="c1">p</font></span>)</span><span class="kw"> such that </span><a NAME="E2:55_1_1_1_3_1"/><span class="lab"><font color="Green" title="E47">L1</font></span>: 
<font color="Maroon" title="c4">xab</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c3">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c5">xa</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="c1">p</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E4:55_1_1_1_3_1"/><span class="lab"><font color="Green" title="E48">L2</font></span>: 
<font color="Maroon" title="c5">xa</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:55_1_1"><span class="lab"><font color="Green" title="E41">P0</font></span></a>, <a class="ref" href="ec_pf_1.html#D3" target="_self" title="EC_PF_1:def.3">QRDef1</a></span>;<br/>
<a NAME="E5:55_1_1_1_3_1"/><span class="lab"><font color="Green" title="E49">L3</font></span>: 
<font color="Maroon" title="c5">xa</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> 2 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <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="c1">p</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:55_1_1_1_3"><span class="lab"><font color="Green" title="E45">A2</font></span></a>, <a class="txt" href="#E4:55_1_1_1_3_1"><span class="lab"><font color="Green" title="E48">L2</font></span></a>, <a class="ref" href="ec_pf_1.html#T11" target="_self" title="EC_PF_1:th.11">XLm2</a></span>;<br/>
<a NAME="E6:55_1_1_1_3_1"/><span class="lab"><font color="Green" title="E50">L4</font></span>: 
<font color="Maroon" title="c5">xa</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1_1_3_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:55_1_1_1_3_1_1"/>
<font color="Maroon" title="c5">xa</font> <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"> contradiction</span><br/>

<a NAME="E2:55_1_1_1_3_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">xa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <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="c1">p</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ec_pf_1.html#T11" target="_self" title="EC_PF_1:th.11">XLm2</a></span>;<br/>
<a NAME="E3:55_1_1_1_3_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">xa</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c5">xa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <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="c1">p</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#T12" title="VECTSP_1:th.12">VECTSP_1:12</a></span>;<br/>
<a NAME="E4:55_1_1_1_3_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">xa</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <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="c1">p</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="ec_pf_1.html#E26"><span class="lab"><font color="Green" title="E21">EXLm4</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:55_1_1_1_3_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:55_1_1_1_3_1"><span class="lab"><font color="Green" title="E48">L2</font></span></a>, <a class="txt" href="#E3:55_1_1_1_3"><span class="lab"><font color="Green" title="E45">A2</font></span></a>, <a class="ref" href="ec_pf_1.html#T11" target="_self" title="EC_PF_1:th.11">XLm2</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/>


</div><span class="kw">end;</span></div>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">xa</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c4">xab</font></span>)</span> <a href="binom.html#K2" title="BINOM:func.2">|^</a> 2 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">xa</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span> <a href="binom.html#K2" title="BINOM:func.2">|^</a> 2</span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c3">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:55_1_1_1_3_1"><span class="lab"><font color="Green" title="E47">L1</font></span></a>, <a class="ref" href="ec_pf_1.html#T28" target="_self" title="EC_PF_1:th.28">EX9</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">xa</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span> <a href="binom.html#K2" title="BINOM:func.2">|^</a> 2</span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">xa</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> 2</span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c3">b</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:55_1_1_1_3_1"><span class="lab"><font color="Green" title="E48">L2</font></span></a>, <a class="ref" href="group_1.html#D3" title="GROUP_1:def.3">GROUP_1:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">xa</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> 2</span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">xa</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> 2</span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c3">b</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ec_pf_1.html#T29" target="_self" title="EC_PF_1:th.29">EX10</a>, <a class="txt" href="#E6:55_1_1_1_3_1"><span class="lab"><font color="Green" title="E50">L4</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p2">(<span class="default"><a href="ec_pf_1.html#NK2" title="EC_PF_1:NK.2">GF</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c3">b</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:55_1_1_1_3_1"><span class="lab"><font color="Green" title="E49">L3</font></span></a>, <a class="ref" href="vectsp_1.html#D10" title="VECTSP_1:def.10">VECTSP_1:def 10</a></span>
<br/>.= 
<font color="Maroon" title="c3">b</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#D8" title="VECTSP_1:def.8">VECTSP_1:def 8</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E8:55_1_1_1_3_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:55_1_1_1_3"><span class="lab"><font color="Green" title="E43">A0</font></span></a>, <a class="ref" href="ec_pf_1.html#D3" target="_self" title="EC_PF_1:def.3">QRDef1</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/>


</div><span class="kw">end;</span></div><a NAME="E6:55_1_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c2">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c3">b</font> is  <a href="ec_pf_1.html#V3" title="EC_PF_1:attr.3">not_quadratic_residue</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:55_1_1_1_3"><span class="lab"><font color="Green" title="E46">A4</font></span></a>, <a class="ref" href="ec_pf_1.html#D4" target="_self" title="EC_PF_1:def.4">QRDef2</a></span>;<br/><span class="kw">hence </span><a NAME="E7:55_1_1_1_3"/>
 <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="c2">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ec_pf_1.html#K2" title="EC_PF_1:func.2">Lege_p</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="ec_pf_1.html#K2" title="EC_PF_1:func.2">Lege_p</a> <font color="Maroon" title="c3">b</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:55_1_1"><span class="lab"><font color="Green" title="E42">P1</font></span></a>, <a class="txt" href="#E2:55_1_1_1_3"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="ref" href="ec_pf_1.html#T35" target="_self" title="EC_PF_1:th.35">QR10</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/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_2"><span class="kw">suppose </span></a><a NAME="E1:55_1_2"/><span class="lab"><font color="Green" title="E41">P2</font></span>: 
 not <font color="Maroon" title="c2">a</font> is  <a href="ec_pf_1.html#V2" title="EC_PF_1:attr.2">quadratic_residue</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="c2">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ec_pf_1.html#K2" title="EC_PF_1:func.2">Lege_p</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="ec_pf_1.html#K2" title="EC_PF_1:func.2">Lege_p</a> <font color="Maroon" title="c3">b</font></span>)</span></span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:55_1_2_1_1"/>
( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c2">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>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:55_1_2_1_1_1"/><span class="lab"><font color="Green" title="E42">P3</font></span>: 
<font color="Maroon" title="c2">a</font> <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="c2">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ec_pf_1.html#K2" title="EC_PF_1:func.2">Lege_p</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="ec_pf_1.html#K2" title="EC_PF_1:func.2">Lege_p</a> <font color="Maroon" title="c3">b</font></span>)</span></span><br/><div class="add"><a NAME="E2:55_1_2_1_1_1"/><span class="kw">then </span>
 <a href="ec_pf_1.html#K2" title="EC_PF_1:func.2">Lege_p</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ec_pf_1.html#T37" target="_self" title="EC_PF_1:th.37">QR12</a></span>;<br/><span class="kw">hence </span><a NAME="E3:55_1_2_1_1_1"/>
 <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="c2">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ec_pf_1.html#K2" title="EC_PF_1:func.2">Lege_p</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="ec_pf_1.html#K2" title="EC_PF_1:func.2">Lege_p</a> <font color="Maroon" title="c3">b</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:55_1_2_1_1_1"><span class="lab"><font color="Green" title="E42">P3</font></span></a>, <a class="ref" href="ec_pf_1.html#T20" target="_self" title="EC_PF_1:th.20">GF8</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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:55_1_2_1_1_2"/><span class="lab"><font color="Green" title="E42">P4</font></span>: 
<font color="Maroon" title="c2">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> 
 ; <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="c2">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ec_pf_1.html#K2" title="EC_PF_1:func.2">Lege_p</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="ec_pf_1.html#K2" title="EC_PF_1:func.2">Lege_p</a> <font color="Maroon" title="c3">b</font></span>)</span></span><br/><div class="add"><a NAME="E2:55_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">P5</font></span>: 
 <a href="ec_pf_1.html#K2" title="EC_PF_1:func.2">Lege_p</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:55_1_2"><span class="lab"><font color="Green" title="E41">P2</font></span></a>, <a class="ref" href="ec_pf_1.html#D5" target="_self" title="EC_PF_1:def.5">QRDef3</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_2_1_1_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:55_1_2_1_1_2_1"/>
( <font color="Maroon" title="c3">b</font> is  <a href="ec_pf_1.html#V2" title="EC_PF_1:attr.2">quadratic_residue</a>  or <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or ( <font color="Maroon" title="c3">b</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;  not <font color="Maroon" title="c3">b</font> is  <a href="ec_pf_1.html#V2" title="EC_PF_1:attr.2">quadratic_residue</a>  ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_2_1_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:55_1_2_1_1_2_1_1"/><span class="lab"><font color="Green" title="E44">A0</font></span>: 
<font color="Maroon" title="c3">b</font> is  <a href="ec_pf_1.html#V2" title="EC_PF_1:attr.2">quadratic_residue</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="c2">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ec_pf_1.html#K2" title="EC_PF_1:func.2">Lege_p</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="ec_pf_1.html#K2" title="EC_PF_1:func.2">Lege_p</a> <font color="Maroon" title="c3">b</font></span>)</span></span><br/><div class="add"><a NAME="E2:55_1_2_1_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A1</font></span>: 
 <a href="ec_pf_1.html#K2" title="EC_PF_1:func.2">Lege_p</a> <font color="Maroon" title="c3">b</font> <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#T36" target="_self" title="EC_PF_1:th.36">QR11</a></span>;<br/><a NAME="E3:55_1_2_1_1_2_1_1"/><span class="lab"><font color="Green" title="E46">A3</font></span>: 
( <font color="Maroon" title="c3">b</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;  ex <font color="Olive" title="b1">x</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="c1">p</font></span>)</span> st <font color="Olive" title="b1">x</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:55_1_2_1_1_2_1_1"><span class="lab"><font color="Green" title="E44">A0</font></span></a>, <a class="ref" href="ec_pf_1.html#D3" target="_self" title="EC_PF_1:def.3">QRDef1</a></span>;<br/><a NAME="E4:55_1_2_1_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A4</font></span>: 
<font color="Maroon" title="c2">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:55_1_2_1_1_2"><span class="lab"><font color="Green" title="E42">P4</font></span></a>, <a class="ref" href="ec_pf_1.html#T20" target="_self" title="EC_PF_1:th.20">GF8</a></span>;<br/><a NAME="E5:55_1_2_1_1_2_1_1"/>
 for <font color="Olive" title="b1">x</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="c1">p</font></span>)</span>  holds  not <font color="Olive" title="b1">x</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c3">b</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_2_1_1_2_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">given </span><font color="Maroon" title="c4">xab</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="c1">p</font></span>)</span><span class="kw"> such that </span><a NAME="E2:55_1_2_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E48">L1</font></span>: 
<font color="Maroon" title="c4">xab</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c3">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c5">xb</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="c1">p</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E4:55_1_2_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E49">L2</font></span>: 
<font color="Maroon" title="c5">xb</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:55_1_2_1_1_2_1_1"><span class="lab"><font color="Green" title="E44">A0</font></span></a>, <a class="ref" href="ec_pf_1.html#D3" target="_self" title="EC_PF_1:def.3">QRDef1</a></span>;<br/>
<a NAME="E5:55_1_2_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E50">L3</font></span>: 
<font color="Maroon" title="c5">xb</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> 2 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <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="c1">p</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:55_1_2_1_1_2_1_1"><span class="lab"><font color="Green" title="E46">A3</font></span></a>, <a class="txt" href="#E4:55_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E49">L2</font></span></a>, <a class="ref" href="ec_pf_1.html#T11" target="_self" title="EC_PF_1:th.11">XLm2</a></span>;<br/>
<a NAME="E6:55_1_2_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E51">L4</font></span>: 
<font color="Maroon" title="c5">xb</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_2_1_1_2_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:55_1_2_1_1_2_1_1_1_1"/>
<font color="Maroon" title="c5">xb</font> <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"> contradiction</span><br/>

<a NAME="E2:55_1_2_1_1_2_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">xb</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <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="c1">p</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ec_pf_1.html#T11" target="_self" title="EC_PF_1:th.11">XLm2</a></span>;<br/>
<a NAME="E3:55_1_2_1_1_2_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">xb</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c5">xb</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <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="c1">p</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#T12" title="VECTSP_1:th.12">VECTSP_1:12</a></span>;<br/>
<a NAME="E4:55_1_2_1_1_2_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">xb</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <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="c1">p</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="ec_pf_1.html#E26"><span class="lab"><font color="Green" title="E21">EXLm4</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:55_1_2_1_1_2_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:55_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E49">L2</font></span></a>, <a class="txt" href="#E3:55_1_2_1_1_2_1_1"><span class="lab"><font color="Green" title="E46">A3</font></span></a>, <a class="ref" href="ec_pf_1.html#T11" target="_self" title="EC_PF_1:th.11">XLm2</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/>


</div><span class="kw">end;</span></div>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">xab</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">xb</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span></span>)</span> <a href="binom.html#K2" title="BINOM:func.2">|^</a> 2 = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">xb</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span> <a href="binom.html#K2" title="BINOM:func.2">|^</a> 2</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:55_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E48">L1</font></span></a>, <a class="ref" href="ec_pf_1.html#T28" target="_self" title="EC_PF_1:th.28">EX9</a></span>
<br/>.= 
<font color="Maroon" title="c2">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">xb</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> 2</span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">xb</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span> <a href="binom.html#K2" title="BINOM:func.2">|^</a> 2</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:55_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E49">L2</font></span></a>, <a class="ref" href="group_1.html#D3" title="GROUP_1:def.3">GROUP_1:def 3</a></span>
<br/>.= 
<font color="Maroon" title="c2">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">xb</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> 2</span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">xb</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> 2</span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ec_pf_1.html#T29" target="_self" title="EC_PF_1:th.29">EX10</a>, <a class="txt" href="#E6:55_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E51">L4</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c2">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p2">(<span class="default"><a href="ec_pf_1.html#NK2" title="EC_PF_1:NK.2">GF</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:55_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E50">L3</font></span></a>, <a class="ref" href="vectsp_1.html#D10" title="VECTSP_1:def.10">VECTSP_1:def 10</a></span>
<br/>.= 
<font color="Maroon" title="c2">a</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#D8" title="VECTSP_1:def.8">VECTSP_1:def 8</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E8:55_1_2_1_1_2_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:55_1_2"><span class="lab"><font color="Green" title="E41">P2</font></span></a>, <a class="txt" href="#E1:55_1_2_1_1_2"><span class="lab"><font color="Green" title="E42">P4</font></span></a>, <a class="ref" href="ec_pf_1.html#D3" target="_self" title="EC_PF_1:def.3">QRDef1</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/>


</div><span class="kw">end;</span></div><a NAME="E6:55_1_2_1_1_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c3">b</font> is  <a href="ec_pf_1.html#V3" title="EC_PF_1:attr.3">not_quadratic_residue</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:55_1_2_1_1_2_1_1"><span class="lab"><font color="Green" title="E47">A4</font></span></a>, <a class="ref" href="ec_pf_1.html#D4" target="_self" title="EC_PF_1:def.4">QRDef2</a></span>;<br/><span class="kw">hence </span><a NAME="E7:55_1_2_1_1_2_1_1"/>
 <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="c2">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ec_pf_1.html#K2" title="EC_PF_1:func.2">Lege_p</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="ec_pf_1.html#K2" title="EC_PF_1:func.2">Lege_p</a> <font color="Maroon" title="c3">b</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:55_1_2_1_1_2"><span class="lab"><font color="Green" title="E43">P5</font></span></a>, <a class="txt" href="#E2:55_1_2_1_1_2_1_1"><span class="lab"><font color="Green" title="E45">A1</font></span></a>, <a class="ref" href="ec_pf_1.html#T35" target="_self" title="EC_PF_1:th.35">QR10</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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_2_1_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:55_1_2_1_1_2_1_2"/><span class="lab"><font color="Green" title="E44">A0</font></span>: 
<font color="Maroon" title="c3">b</font> <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="c2">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ec_pf_1.html#K2" title="EC_PF_1:func.2">Lege_p</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="ec_pf_1.html#K2" title="EC_PF_1:func.2">Lege_p</a> <font color="Maroon" title="c3">b</font></span>)</span></span><br/><div class="add"><a NAME="E2:55_1_2_1_1_2_1_2"/><span class="kw">then </span>
 <a href="ec_pf_1.html#K2" title="EC_PF_1:func.2">Lege_p</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ec_pf_1.html#T37" target="_self" title="EC_PF_1:th.37">QR12</a></span>;<br/><span class="kw">hence </span><a NAME="E3:55_1_2_1_1_2_1_2"/>
 <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="c2">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ec_pf_1.html#K2" title="EC_PF_1:func.2">Lege_p</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="ec_pf_1.html#K2" title="EC_PF_1:func.2">Lege_p</a> <font color="Maroon" title="c3">b</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:55_1_2_1_1_2_1_2"><span class="lab"><font color="Green" title="E44">A0</font></span></a>, <a class="ref" href="ec_pf_1.html#T20" target="_self" title="EC_PF_1:th.20">GF8</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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_2_1_1_2_1_3"><span class="kw">suppose </span></a><a NAME="E1:55_1_2_1_1_2_1_3"/><span class="lab"><font color="Green" title="E44">A0</font></span>: 
( <font color="Maroon" title="c3">b</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;  not <font color="Maroon" title="c3">b</font> is  <a href="ec_pf_1.html#V2" title="EC_PF_1:attr.2">quadratic_residue</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="c2">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ec_pf_1.html#K2" title="EC_PF_1:func.2">Lege_p</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="ec_pf_1.html#K2" title="EC_PF_1:func.2">Lege_p</a> <font color="Maroon" title="c3">b</font></span>)</span></span><br/><div class="add"><a NAME="E2:55_1_2_1_1_2_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A1</font></span>: 
 <a href="ec_pf_1.html#K2" title="EC_PF_1:func.2">Lege_p</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.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>;<br/><a NAME="E3:55_1_2_1_1_2_1_3"/><span class="lab"><font color="Green" title="E46">A4</font></span>: 
<font color="Maroon" title="c2">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:55_1_2_1_1_2"><span class="lab"><font color="Green" title="E42">P4</font></span></a>, <a class="txt" href="#E1:55_1_2_1_1_2_1_3"><span class="lab"><font color="Green" title="E44">A0</font></span></a>, <a class="ref" href="ec_pf_1.html#T20" target="_self" title="EC_PF_1:th.20">GF8</a></span>;<br/><a NAME="E4:55_1_2_1_1_2_1_3"/>
 ex <font color="Olive" title="b1">x</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="c1">p</font></span>)</span> st <font color="Olive" title="b1">x</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c3">b</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_2_1_1_2_1_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">consider </span><font color="Maroon" title="c4">g</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="c1">p</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E2:55_1_2_1_1_2_1_3_1"/><span class="lab"><font color="Green" title="E47">XP1</font></span>: 
 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="c1">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="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <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="c1">p</font></span>)</span> holds <br/> ex <font color="Olive" title="b2">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Olive" title="b2">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ec_pf_1.html#T33" target="_self" title="EC_PF_1:th.33">LMCycle5</a></span>;<br/>
<a NAME="E3:55_1_2_1_1_2_1_3_1"/>
<font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <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="c1">p</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:55_1_2_1_1_2"><span class="lab"><font color="Green" title="E42">P4</font></span></a>, <a class="ref" href="ec_pf_1.html#T11" target="_self" title="EC_PF_1:th.11">XLm2</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">na</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E5:55_1_2_1_1_2_1_3_1"/><span class="lab"><font color="Green" title="E48">XP2</font></span>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Maroon" title="c5">na</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:55_1_2_1_1_2_1_3_1"><span class="lab"><font color="Green" title="E47">XP1</font></span></a></span>;<br/>
<a NAME="E6:55_1_2_1_1_2_1_3_1"/>
<font color="Maroon" title="c3">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <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="c1">p</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:55_1_2_1_1_2_1_3"><span class="lab"><font color="Green" title="E44">A0</font></span></a>, <a class="ref" href="ec_pf_1.html#T11" target="_self" title="EC_PF_1:th.11">XLm2</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">nb</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E8:55_1_2_1_1_2_1_3_1"/><span class="lab"><font color="Green" title="E49">XP3</font></span>: 
<font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Maroon" title="c6">nb</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:55_1_2_1_1_2_1_3_1"><span class="lab"><font color="Green" title="E47">XP1</font></span></a></span>;<br/>
<a NAME="E9:55_1_2_1_1_2_1_3_1"/><span class="lab"><font color="Green" title="E50">XP4</font></span>: 
<font color="Maroon" title="c5">na</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">na</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="c5">na</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="nat_d.html#T2" title="NAT_D:th.2">NAT_D:2</a></span>;<br/>
<a NAME="E10:55_1_2_1_1_2_1_3_1"/><span class="lab"><font color="Green" title="E51">XP5</font></span>: 
<font color="Maroon" title="c6">nb</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">nb</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="c6">nb</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="nat_d.html#T2" title="NAT_D:th.2">NAT_D:2</a></span>;<br/>
<a NAME="E11:55_1_2_1_1_2_1_3_1"/>
<font color="Maroon" title="c5">na</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_2_1_1_2_1_3_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:55_1_2_1_1_2_1_3_1_1"/><span class="lab"><font color="Green" title="E52">XP61</font></span>: 
<font color="Maroon" title="c5">na</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"> contradiction</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c7">nn</font> = <font color="Maroon" title="c5">na</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2 as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<a NAME="E3:55_1_2_1_1_2_1_3_1_1"/>
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Maroon" title="c7">nn</font></span>)</span> <a href="binom.html#K2" title="BINOM:func.2">|^</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:55_1_2_1_1_2_1_3_1"><span class="lab"><font color="Green" title="E48">XP2</font></span></a>, <a class="txt" href="#E9:55_1_2_1_1_2_1_3_1"><span class="lab"><font color="Green" title="E50">XP4</font></span></a>, <a class="txt" href="#E1:55_1_2_1_1_2_1_3_1_1"><span class="lab"><font color="Green" title="E52">XP61</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="E4:55_1_2_1_1_2_1_3_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:55_1_2"><span class="lab"><font color="Green" title="E41">P2</font></span></a>, <a class="txt" href="#E1:55_1_2_1_1_2"><span class="lab"><font color="Green" title="E42">P4</font></span></a>, <a class="ref" href="ec_pf_1.html#D3" target="_self" title="EC_PF_1:def.3">QRDef1</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/>


</div><span class="kw">end;</span></div>
<a NAME="E12:55_1_2_1_1_2_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">XP7</font></span>: 
<font color="Maroon" title="c5">na</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> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T12" title="NAT_D:th.12">NAT_D:12</a></span>;<br/>
<a NAME="E13:55_1_2_1_1_2_1_3_1"/>
<font color="Maroon" title="c6">nb</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_2_1_1_2_1_3_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:55_1_2_1_1_2_1_3_1_2"/><span class="lab"><font color="Green" title="E53">XP61</font></span>: 
<font color="Maroon" title="c6">nb</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"> contradiction</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c7">nn</font> = <font color="Maroon" title="c6">nb</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2 as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<a NAME="E3:55_1_2_1_1_2_1_3_1_2"/>
<font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Maroon" title="c7">nn</font></span>)</span> <a href="binom.html#K2" title="BINOM:func.2">|^</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:55_1_2_1_1_2_1_3_1"><span class="lab"><font color="Green" title="E49">XP3</font></span></a>, <a class="txt" href="#E10:55_1_2_1_1_2_1_3_1"><span class="lab"><font color="Green" title="E51">XP5</font></span></a>, <a class="txt" href="#E1:55_1_2_1_1_2_1_3_1_2"><span class="lab"><font color="Green" title="E53">XP61</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="E4:55_1_2_1_1_2_1_3_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:55_1_2_1_1_2_1_3"><span class="lab"><font color="Green" title="E44">A0</font></span></a>, <a class="ref" href="ec_pf_1.html#D3" target="_self" title="EC_PF_1:def.3">QRDef1</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/>


</div><span class="kw">end;</span></div>
<a NAME="E14:55_1_2_1_1_2_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">XP9</font></span>: 
<font color="Maroon" title="c6">nb</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> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T12" title="NAT_D:th.12">NAT_D:12</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">nc</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">na</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">nb</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> ;<br/>
<span class="lab"><font color="Green" title="E54">XP10</font></span>: <font color="Maroon" title="c5">na</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c6">nb</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">na</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> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c6">nb</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:55_1_2_1_1_2_1_3_1"><span class="lab"><font color="Green" title="E52">XP7</font></span></a>, <a class="ref" href="nat_d.html#T2" title="NAT_D:th.2">NAT_D:2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">na</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> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">nb</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> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:55_1_2_1_1_2_1_3_1"><span class="lab"><font color="Green" title="E53">XP9</font></span></a>, <a class="ref" href="nat_d.html#T2" title="NAT_D:th.2">NAT_D:2</a></span>
<br/>.= 
<font color="Maroon" title="c7">nc</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> 2

;<br/>
<font color="Maroon" title="c2">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c3">b</font> = 
<font color="Maroon" title="c4">g</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">na</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c6">nb</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:55_1_2_1_1_2_1_3_1"><span class="lab"><font color="Green" title="E48">XP2</font></span></a>, <a class="txt" href="#E8:55_1_2_1_1_2_1_3_1"><span class="lab"><font color="Green" title="E49">XP3</font></span></a>, <a class="ref" href="ec_pf_1.html#T30" target="_self" title="EC_PF_1:th.30">EX11</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Maroon" title="c7">nc</font></span>)</span> <a href="binom.html#K2" title="BINOM:func.2">|^</a> 2
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:55_1_2_1_1_2_1_3_1"><span class="lab"><font color="Green" title="E54">XP10</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="E18:55_1_2_1_1_2_1_3_1"/>
 ex <font color="Olive" title="b1">x</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="c1">p</font></span>)</span> st <font color="Olive" title="b1">x</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c3">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E5:55_1_2_1_1_2_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c2">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c3">b</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="#E3:55_1_2_1_1_2_1_3"><span class="lab"><font color="Green" title="E46">A4</font></span></a>, <a class="ref" href="ec_pf_1.html#D3" target="_self" title="EC_PF_1:def.3">QRDef1</a></span>;<br/><span class="kw">hence </span><a NAME="E6:55_1_2_1_1_2_1_3"/>
 <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="c2">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ec_pf_1.html#K2" title="EC_PF_1:func.2">Lege_p</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="ec_pf_1.html#K2" title="EC_PF_1:func.2">Lege_p</a> <font color="Maroon" title="c3">b</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:55_1_2_1_1_2"><span class="lab"><font color="Green" title="E43">P5</font></span></a>, <a class="txt" href="#E2:55_1_2_1_1_2_1_3"><span class="lab"><font color="Green" title="E45">A1</font></span></a>, <a class="ref" href="ec_pf_1.html#T36" target="_self" title="EC_PF_1:th.36">QR11</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/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:55_1_2"/>
 <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="c2">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ec_pf_1.html#K2" title="EC_PF_1:func.2">Lege_p</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="ec_pf_1.html#K2" title="EC_PF_1:func.2">Lege_p</a> <font color="Maroon" title="c3">b</font></span>)</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/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
