let X, Y be non empty compact Subset of (TOP-REAL 2); :: thesis: ( X c= Y & ( N-min Y in X or N-max Y in X ) implies N-bound X = N-bound Y )
assume that
A1: X c= Y and
A2: ( N-min Y in X or N-max Y in X ) ; :: thesis: N-bound X = N-bound Y
A3: (N-max X) `2 = N-bound X by EUCLID:52;
A4: (N-max Y) `2 = N-bound Y by EUCLID:52;
A5: (N-min Y) `2 = N-bound Y by EUCLID:52;
(N-min X) `2 = N-bound X by EUCLID:52;
hence N-bound X = N-bound Y by A1, A2, A3, A5, A4, Th15, Th16; :: thesis: verum