reserve i,i1,i2,i3,i4,i5,j,r,a,b,x,y for Integer,
  d,e,k,n for Nat,
  fp,fk for FinSequence of INT,
  f,f1,f2 for FinSequence of REAL,
  p for Prime;
reserve fr for FinSequence of REAL;
reserve fr,f for FinSequence of INT;
reserve b,m for Nat;
reserve b for Integer;
reserve m for Integer;
reserve fp for FinSequence of NAT;
reserve a,m for Nat;
reserve f,g,h,k for FinSequence of REAL;
reserve X for finite set,
  F for FinSequence of bool X;
reserve q for Prime;

theorem
  p>2 & q>2 & p<>q & p mod 4 = 3 & q mod 4 = 3 implies Lege(p,q) = -Lege (q,p)
proof
  assume that
A1: p>2 and
A2: q>2 and
A3: p<>q and
A4: p mod 4 = 3 and
A5: q mod 4 = 3;
  q>1 by INT_2:def 4;
  then
A6: q-'1 = q-1 by XREAL_1:233;
  q = 4*(q div 4)+3 by A5,NAT_D:2;
  then q-'1 = 2*(2*(q div 4)+1) by A6;
  then
A7: (q-'1) div 2=2*(q div 4)+1;
  p>1 by INT_2:def 4;
  then
A8: p-'1 = p-1 by XREAL_1:233;
  p=4*(p div 4)+3 by A4,NAT_D:2;
  then p-'1 = 2*(2*(p div 4)+1) by A8;
  then (p-'1) div 2=2*(p div 4)+1;
  then
A9: Lege(p,q)*Lege(q,p) =(-1)|^((2*(p div 4)+1)*(2*(q div 4)+1)) by A1,A2,A3,A7
,Th49
    .=((-1)|^(2*(p div 4)+1))|^(2*(q div 4)+1) by NEWTON:9
    .= ((-1)|^(2*(p div 4))*(-1))|^(2*(q div 4)+1) by NEWTON:6
    .= ((((-1)|^2)|^(p div 4))*(-1))|^(2*(q div 4)+1) by NEWTON:9
    .= ((1|^2)|^(p div 4)*(-1))|^(2*(q div 4)+1) by WSIERP_1:1
    .= (-1)|^(2*(q div 4))*(-1) by NEWTON:6
    .= ((-1)|^2)|^(q div 4) * (-1) by NEWTON:9
    .= (1|^2)|^(q div 4) *(-1) by WSIERP_1:1
    .= 1*(-1);
  per cases by Th25;
  suppose
    Lege(p,q) = 1;
    hence thesis by A9;
  end;
  suppose
    Lege(p,q) = 0;
    hence thesis by A9;
  end;
  suppose
    Lege(p,q) = -1;
    hence thesis by A9;
  end;
end;
