theorem :: POLYEQ_4:17

for x, y, a, b being Real
for n being Element of NAT st (x |^ n) + (y |^ n) = a & (x |^ n) - (y |^ n) = b & n is even & n >= 1 & a + b > 0 & a - b > 0 & not ( x = n -root ((a + b) / 2) & y = n -root ((a - b) / 2) ) & not ( x = n -root ((a + b) / 2) & y = - (n -root ((a - b) / 2)) ) & not ( x = - (n -root ((a + b) / 2)) & y = n -root ((a - b) / 2) ) holds
( x = - (n -root ((a + b) / 2)) & y = - (n -root ((a - b) / 2)) )