reserve a, a9, a1, a2, a3, b, b9, c, c9, d, d9, h, p, q, x, x1, x2, x3, u, v,
  y, z for Real;

theorem Th12:
  (for x holds Polynom(a,b,c,d,x) = Polynom(a9,b9,c9,d9,x))
  implies a = a9 & b = b9 & c = c9 & d = d9
proof
  (-1) |^ 3 = - (1 |^ (2+1)) by Lm1,POWER:2
    .= - ((1 |^ 2)*1);
  then
A1: (0 |^ 3) = 0 & (-1) |^ 3 = - 1 by NEWTON:11;
A2: 2 |^ 3 = 2 |^ (2+1) .= (2 |^ (1+1))*2 by NEWTON:6
    .= ((2 |^ 1)*2)*2 by NEWTON:6
    .= (2 |^ 1)*(2*2);
  assume
A3: for x holds Polynom(a,b,c,d,x) = Polynom(a9,b9,c9,d9,x);
  then
A4: Polynom(a,b,c,d,2) = Polynom(a9,b9,c9,d9,2);
  Polynom(a,b,c,d,1) = Polynom(a9,b9,c9,d9,1) by A3;
  then a*1 + b*1 +c*1 +d = Polynom(a9,b9,c9,d9,1);
  then
A5: a + b +c +d = a9*1+ b9*1 +c9*1 +d9
    .= a9+ b9 +c9 +d9;
  Polynom(a,b,c,d,0) = Polynom(a9,b9,c9,d9,0) & Polynom(a,b,c,d,-1) =
  Polynom( a9,b9,c9,d9,-1) by A3;
  hence thesis by A5,A1,A4,A2;
end;
