reserve x,y,a,b,c,p,q for Real;
reserve m,n for Element of NAT;

theorem
  a <> 0 & b/a<0 & c/a>0 & n is even & n >= 1 & delta(a,b,c) >= 0 &
    Polynom(a,b,c,x|^ n) = 0 implies
  x = n-root((-b+sqrt delta(a,b,c))/(2*a)) or
  x = -n-root((-b+sqrt delta(a,b,c))/(2*a)) or
  x = n-root((-b-sqrt delta(a,b,c))/(2*a)) or
  x = -n-root((-b-sqrt delta(a,b,c))/(2*a))
proof
  assume that
A1: a <>0 and
A2: b/a<0 & c/a>0 & n is even & n >= 1 and
A3: delta(a,b,c)>=0 and
A4: Polynom(a,b,c,x|^ n)=0;
:: theorem Th4:
::   a>0 & n is even & n >= 1 & x |^ n = a implies x = n-root a or x = -n-root a
  now
    per cases by A1,A3,A4,POLYEQ_1:5;
    suppose
      x|^ n = (-b+sqrt delta(a,b,c))/(2*a);
      then x = n-root((-b+sqrt delta(a,b,c))/(2*a)) or
           x = -n-root((-b+sqrt delta(a,b,c))/(2*a)) by A2,A3,Th1,Th4;
      hence thesis;
    end;
    suppose
      x|^ n = (-b-sqrt delta(a,b,c))/(2*a);
      then x = n-root((-b-sqrt delta(a,b,c))/(2*a)) or
         x = -n-root((-b-sqrt delta(a,b,c))/(2*a)) by A2,A3,Th1,Th4;
      hence thesis;
    end;
  end;
  hence thesis;
end;
