
theorem Th9:
  for a1,a2,a3,a4,a5,b1,b2,b3,b4,b5 being Real st (for x
being Real holds Polynom(a1,a2,a3,a4,a5,x) = Polynom(b1,b2,b3,b4,b5,x))
  holds a1 = b1 & a2 = b2 & a3 = b3 & a4 = b4 & a5 = b5
proof
A1: (-2)|^ 3 = (-2)^2*(-2) by Th4
    .= -(4*2);
A2: (-2)|^ 4 = 16 by Lm1,POWER:1,62;
  let a1,a2,a3,a4,a5,b1,b2,b3,b4,b5 be Real;
  assume
A3: for x being Real holds Polynom(a1,a2,a3,a4,a5,x) = Polynom(b1
  ,b2,b3,b4,b5,x);
  then
A4: Polynom(a1,a2,a3,a4,a5,-2) = Polynom(b1,b2,b3,b4,b5,-2);
A5: a5 = b5 & Polynom(a1,a2,a3,a4,a5,2) = Polynom(b1,b2,b3,b4,b5,2) by A3,Th7;
  a1-b1 = b3-a3 & a2-b2 = b4-a4 by A3,Th8;
  hence thesis by A5,A4,A2,A1,POWER:61,62;
end;
