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
  a <> 0 & (for x holds Polynom(a,b,c,d,x) = Tri(a,x1,x2,x3,x)) implies
  b/a = -(x1+x2+x3) & c/a = x1*x2 +x2*x3 +x1*x3 & d/a = -x1*x2*x3
proof
  assume
A1: a <> 0;
  set t3 = d/a;
  set t2 = c/a;
  set t1 = b/a;
  set d9 = -x1*x2*x3;
  set c9 = x1*x2+x2*x3+x1*x3;
  set b9 = -(x1+x2+x3);
  assume
A2: for x holds Polynom(a,b,c,d,x) = Tri(a,x1,x2,x3,x);
  now
    let x;
    set t = a*(x |^ 3)+ b*x^2 +c*x +d;
    set r8 = ((x-x1)*(x-x2)*(x-x3));
    x |^ 3 = x |^ (2+1) .= (x |^ (1+1))*x by NEWTON:6
      .= ((x |^ 1)*x)*x by NEWTON:6;
    then
A3: x |^ 3 = (x*x)*x;
    Polynom(a,b,c,d,x) = Tri(a,x1,x2,x3,x)by A2;
    then
A4: t/a = r8 by A1,XCMPLX_1:89;
    a"*t = (a"*a)*(x |^ 3)+ (a"*b)*x^2 +a"*(c*x +d)
      .= 1*(x |^ 3)+ (a"*b)*x^2 +((a"*c)*x +a"*d) by A1,XCMPLX_0:def 7
      .= 1*(x |^ 3)+ t1*x^2 +((a"*c)*x +a"*d) by XCMPLX_0:def 9
      .= 1*(x |^ 3)+ t1*x^2 + (t2*x +a"*d) by XCMPLX_0:def 9
      .= 1*(x |^ 3)+ t1*x^2 + (t2*x + t3) by XCMPLX_0:def 9
      .= Polynom(1,t1,t2,t3,x);
    hence Polynom(1,t1,t2,t3,x) = Polynom(1,b9, c9,d9,x) by A4,A3,
XCMPLX_0:def 9;
  end;
  hence thesis by Th12;
end;
