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

theorem
  a <> 0 & b/a<0 & n is even & n >= 1 & Polynom(a,b,0,x|^ n) = 0
  implies x=0 or x = n-root -(b/a) or x = -n-root -(b/a)
proof
  assume that
A1: a<>0 and
A2: b/a<0 and
A3: n is even & n >= 1 and
A4: Polynom(a,b,0,x|^ n) = 0;
A5: -(b/a)>0 by A2,XREAL_1:58;
  now
    per cases by A1,A4,Th5;
    suppose
      x|^ n = 0;
      hence thesis by PREPOWER:5;
    end;
    suppose
      x|^ n = -(b/a);
      hence thesis by A3,A5,Th4;
    end;
  end;
  hence thesis;
end;
