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;

theorem Th11:
  i is_quadratic_residue_mod m & i,j are_congruent_mod m implies
  j is_quadratic_residue_mod m
proof
  assume that
A1: i is_quadratic_residue_mod m and
A2: i,j are_congruent_mod m;
  consider x being Integer such that
A3: (x^2 - i) mod m = 0 by A1;
A4: (i - j) mod m = 0 by Lm1,A2;
  (x^2 - j) mod m = ((x^2 - i) + (i - j)) mod m
    .= (((x^2 - i) mod m) + ((i - j) mod m)) mod m by NAT_D:66
    .= 0 by A3,A4,NAT_D:65;
  hence thesis;
end;
