
theorem Th7:
  for a1,a2,a3,a4,a5,b1,b2,b3,b4,b5 being Real holds (for x
being Real holds Polynom(a1,a2,a3,a4,a5,x) = Polynom(b1,b2,b3,b4,b5,x))
  implies a5=b5 & a1-a2+a3-a4 = b1-b2+b3-b4 & a1+a2+a3+a4 = b1+b2+b3+b4
proof
  set x=-1;
  let a1,a2,a3,a4,a5,b1,b2,b3,b4,b5 be Real;
A1: 0|^ 3 =0 & 0|^ 4 = 0 by NEWTON:11;
  assume
A2: for x being Real holds Polynom(a1,a2,a3,a4,a5,x) = Polynom(b1
  ,b2,b3,b4,b5,x);
  then
A3: Polynom(a1,a2,a3,a4,a5,-1) = Polynom(b1,b2,b3,b4,b5,-1);
A5: x|^ 3=x^2*x & (x|^ 3)*x=x|^ 4 by Th4;
  Polynom(a1,a2,a3,a4,a5,0) = Polynom(b1,b2,b3,b4,b5,0) & Polynom(a1,a2,a3
  ,a4, a5,1) = Polynom(b1,b2,b3,b4,b5,1) by A2;
  hence thesis by A1,A3,A5;
end;
