
theorem Th15:
  for x being Integer
  holds x|^2, 0 are_congruent_mod 3 or x|^2, 1 are_congruent_mod 3
proof
let x be Integer;
x,0 are_congruent_mod 3 or ... or x,(3-1) are_congruent_mod 3 by Th13;
then A1:
x,0 are_congruent_mod 3 or ... or x,2 are_congruent_mod 3;
per cases by A1;
suppose x,0 are_congruent_mod 3;
  then x*x, 0*0 are_congruent_mod 3 by INT_1:18;
  hence thesis by NEWTON:81;
  end;
suppose x,1 are_congruent_mod 3;
  then x*x, 1*1 are_congruent_mod 3 by INT_1:18;
  hence thesis by NEWTON:81;
  end;
suppose x,2 are_congruent_mod 3;
  then x*x, 2*2 are_congruent_mod 3 by INT_1:18;
  then 4, x*x are_congruent_mod 3 by INT_1:14;
  then (4-3),x*x are_congruent_mod 3 by INT_1:22;
  then x*x, (4-3) are_congruent_mod 3 by INT_1:14;
  hence thesis by NEWTON:81;
  end;
end;
