reserve n for Nat;

theorem Th21:
  for X,Y be non empty compact Subset of TOP-REAL 2 st X c= Y &
  W-min Y in X holds W-min X = W-min Y
proof
  let X,Y be non empty compact Subset of TOP-REAL 2;
  assume that
A1: X c= Y and
A2: W-min Y in X;
A3: W-bound X <= (W-min Y)`1 by A2,PSCOMP_1:24;
A4: (W-min X)`1 = W-bound X by EUCLID:52;
A5: (W-min Y)`1 = W-bound Y by EUCLID:52;
A6: W-bound X >= W-bound Y by A1,PSCOMP_1:69;
  then
A7: W-bound X = W-bound Y by A5,A3,XXREAL_0:1;
  W-min Y in W-most X by A2,A6,A5,A3,SPRECT_2:12,XXREAL_0:1;
  then
A8: (W-min X)`2 <= (W-min Y)`2 by PSCOMP_1:31;
  W-min X in X by SPRECT_1:13;
  then W-min X in W-most Y by A1,A6,A4,A5,A3,SPRECT_2:12,XXREAL_0:1;
  then (W-min X)`2 >= (W-min Y)`2 by PSCOMP_1:31;
  then (W-min X)`2 = (W-min Y)`2 by A8,XXREAL_0:1;
  hence thesis by A4,A5,A7,TOPREAL3:6;
end;
