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;

theorem
  a is_quadratic_residue_mod p & b is_quadratic_residue_mod p implies
  a*b is_quadratic_residue_mod p
proof
  assume that
A1: a is_quadratic_residue_mod p and
A2: b is_quadratic_residue_mod p;
  consider i being Integer such that
A3: (i^2 - a) mod p =0 by A1;
  consider j being Integer such that
A4: (j^2 - b) mod p =0 by A2;
A5: j^2,b are_congruent_mod p by A4,INT_1:62;
  i^2,a are_congruent_mod p by A3,INT_1:62;
  then (i^2)*(j^2),a*b are_congruent_mod p by A5,INT_1:18;
  then ((i*j)^2 - a*b) mod p = 0 by INT_1:62;
  hence thesis;
end;
