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

theorem
  (x|^ n)+(y|^ n)=p & (x|^ n)*(y|^ n)=q & p^2-4*q>=0 & n is odd
  implies x=n-root((p+sqrt(p^2-4*q))/2) & y=n-root((p-sqrt(p^2-4*q))/2) or x=n
  -root((p-sqrt(p^2-4*q))/2) & y=n-root((p+sqrt(p^2-4*q))/2)
proof
  assume that
A1: (x|^ n)+(y|^ n)=p & (x|^ n)*(y|^ n)=q & p^2-4*q>=0 and
A2: n is odd;
  x|^ n=(p+sqrt(p^2-4*q))/2 & y|^ n=(p-sqrt(p^2-4*q))/2 or x|^ n=(p-sqrt(p
  ^2-4*q))/2 & y|^ n=(p+sqrt(p^2-4*q))/2 by A1,Th14;
  hence thesis by A2,POWER:4;
end;
