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

theorem
  x|^ n+y|^ n=a & x|^ n-y|^ n=b & n is even & n >= 1 & a+b>0 & a-b>0
implies x=n-root((a+b)/2) & y=n-root((a-b)/2) or x=n-root((a+b)/2) & y=-n-root(
(a-b)/2) or x=-n-root((a+b)/2)& y=n-root((a-b)/2) or x=-n-root((a+b)/2) & y=-n
  -root((a-b)/2)
proof
  assume
A1: x|^ n+y|^ n=a & x|^ n-y|^ n=b;
  assume that
A2: n is even & n >= 1 and
A3: a+b>0 & a-b>0;
  (a+b)/2>0 & (a-b)/2>0 by A3,XREAL_1:139;
  hence thesis by A1,A2,Th4;
end;
