reserve a,b for Complex;
reserve z for Complex;
reserve n0 for non zero Nat;
reserve a0,a1,a2,s1,s2 for Complex;
reserve a3,x,q,r,s,s3 for Complex;

theorem Th17:
  1_root_of_cubic(a0,a1,a2)+2_root_of_cubic(a0,a1,a2)+
  3_root_of_cubic(a0,a1,a2) = -a2
proof
  per cases;
  suppose
A1: 3*a1 - a2|^2 = 0;
    then
A2: ex r,s1 st r = (9*a2*a1 - 2*a2|^3 - 27*a0)/54 & s1 = 3-root(2*r) &
    3_root_of_cubic(a0,a1,a2) = -s1/2-a2/3-s1*(2-root 3)*<i>/2 by Def4;
    ( ex r,s1 st r = (9*a2*a1 - 2*a2|^3 - 27*a0)/54 & s1 = 3-root(2*r) &
1_root_of_cubic(a0,a1,a2) = s1-a2/3)& ex r,s1 st r = (9*a2*a1 - 2*a2|^3 - 27*
a0)/54 & s1 = 3-root(2*r) & 2_root_of_cubic(a0,a1,a2) = -s1/2-a2/3+s1*(2-root 3
    )*<i>/2 by A1,Def2,Def3;
    hence thesis by A2;
  end;
  suppose
A3: 3*a1 - a2|^2 <> 0;
    then
A4: ex q,r,s,s1,s2 st q = (3*a1 - a2|^2)/9 & r = (9*a2*a1 - 2*a2|^3 - 27*
    a0)/54 & s = 2-root(q|^3+r|^2) & s1 = 3-root(r+s) & s2 = -q/s1 &
3_root_of_cubic(a0,a1,a2) = -(s1+s2)/2-a2/3-(s1-s2)*(2-root 3)*<i>/2 by Def4;
    ( ex q,r,s,s1,s2 st q = (3*a1 - a2|^2)/9 & r = (9*a2*a1 - 2*a2|^3 - 27
    *a0)/54 & s = 2-root(q|^3+r|^2) & s1 = 3-root(r+s) & s2 = -q/s1 &
1_root_of_cubic(a0,a1, a2) = s1+s2-a2/3)& ex q,r,s,s1,s2 st q = (3*a1 - a2|^2)/
9 & r = (9*a2*a1 - 2* a2|^3 - 27*a0)/54 & s = 2-root(q|^3+r|^2) & s1 = 3-root(r
+s) & s2 = -q/s1 & 2_root_of_cubic(a0,a1,a2) = -(s1+s2)/2-a2/3+(s1-s2)*(2-root
    3)*<i>/2 by A3,Def2,Def3;
    hence thesis by A4;
  end;
end;
