reserve n for Nat;

theorem Th16:
  for X,Y be non empty compact Subset of TOP-REAL 2 st X c= Y &
  N-max Y in X holds N-max X = N-max Y
proof
  let X,Y be non empty compact Subset of TOP-REAL 2;
  assume that
A1: X c= Y and
A2: N-max Y in X;
A3: N-bound X >= (N-max Y)`2 by A2,PSCOMP_1:24;
A4: (N-max X)`2 = N-bound X by EUCLID:52;
A5: (N-max Y)`2 = N-bound Y by EUCLID:52;
A6: N-bound X <= N-bound Y by A1,PSCOMP_1:66;
  then
A7: N-bound X = N-bound Y by A5,A3,XXREAL_0:1;
  N-max Y in N-most X by A2,A6,A5,A3,SPRECT_2:10,XXREAL_0:1;
  then
A8: (N-max X)`1 >= (N-max Y)`1 by PSCOMP_1:39;
  N-max X in X by SPRECT_1:11;
  then N-max X in N-most Y by A1,A6,A4,A5,A3,SPRECT_2:10,XXREAL_0:1;
  then (N-max X)`1 <= (N-max Y)`1 by PSCOMP_1:39;
  then (N-max X)`1 = (N-max Y)`1 by A8,XXREAL_0:1;
  hence thesis by A4,A5,A7,TOPREAL3:6;
end;
