reserve n for Nat;

theorem Th17:
  for X,Y be non empty compact Subset of TOP-REAL 2 st X c= Y &
  E-min Y in X holds E-min X = E-min Y
proof
  let X,Y be non empty compact Subset of TOP-REAL 2;
  assume that
A1: X c= Y and
A2: E-min Y in X;
A3: E-bound X >= (E-min Y)`1 by A2,PSCOMP_1:24;
A4: (E-min X)`1 = E-bound X by EUCLID:52;
A5: (E-min Y)`1 = E-bound Y by EUCLID:52;
A6: E-bound X <= E-bound Y by A1,PSCOMP_1:67;
  then
A7: E-bound X = E-bound Y by A5,A3,XXREAL_0:1;
  E-min Y in E-most X by A2,A6,A5,A3,SPRECT_2:13,XXREAL_0:1;
  then
A8: (E-min X)`2 <= (E-min Y)`2 by PSCOMP_1:47;
  E-min X in X by SPRECT_1:14;
  then E-min X in E-most Y by A1,A6,A4,A5,A3,SPRECT_2:13,XXREAL_0:1;
  then (E-min X)`2 >= (E-min Y)`2 by PSCOMP_1:47;
  then (E-min X)`2 = (E-min Y)`2 by A8,XXREAL_0:1;
  hence thesis by A4,A5,A7,TOPREAL3:6;
end;
