reserve n for Nat;

theorem Th20:
  for X,Y be non empty compact Subset of TOP-REAL 2 st X c= Y &
  S-max Y in X holds S-max X = S-max Y
proof
  let X,Y be non empty compact Subset of TOP-REAL 2;
  assume that
A1: X c= Y and
A2: S-max Y in X;
A3: S-bound X <= (S-max Y)`2 by A2,PSCOMP_1:24;
A4: (S-max X)`2 = S-bound X by EUCLID:52;
A5: (S-max Y)`2 = S-bound Y by EUCLID:52;
A6: S-bound X >= S-bound Y by A1,PSCOMP_1:68;
  then
A7: S-bound X = S-bound Y by A5,A3,XXREAL_0:1;
  S-max Y in S-most X by A2,A6,A5,A3,SPRECT_2:11,XXREAL_0:1;
  then
A8: (S-max X)`1 >= (S-max Y)`1 by PSCOMP_1:55;
  S-max X in X by SPRECT_1:12;
  then S-max X in S-most Y by A1,A6,A4,A5,A3,SPRECT_2:11,XXREAL_0:1;
  then (S-max X)`1 <= (S-max Y)`1 by PSCOMP_1:55;
  then (S-max X)`1 = (S-max Y)`1 by A8,XXREAL_0:1;
  hence thesis by A4,A5,A7,TOPREAL3:6;
end;
