let X, Y be non empty compact Subset of (TOP-REAL 2); :: thesis: ( X c= Y & ( W-min Y in X or W-max Y in X ) implies W-bound X = W-bound Y )
assume A1: ( X c= Y & ( W-min Y in X or W-max Y in X ) ) ; :: thesis: W-bound X = W-bound Y
( (W-min X) `1 = W-bound X & (W-max X) `1 = W-bound X & (W-min Y) `1 = W-bound Y & (W-max Y) `1 = W-bound Y ) by EUCLID:56;
hence W-bound X = W-bound Y by A1, Th21, Th22; :: thesis: verum