reserve n for Nat;

theorem
  for X,Y be non empty compact Subset of TOP-REAL 2 st S-bound X <
  S-bound Y holds S-min (X\/Y) = S-min X
proof
  let X,Y be non empty compact Subset of TOP-REAL 2;
A1: (S-min(X\/Y))`2 = S-bound (X\/Y) by EUCLID:52;
A2: X\/Y is compact by COMPTS_1:10;
  then
A3: S-min(X\/Y) in X\/Y by SPRECT_1:12;
A4: S-min X in X by SPRECT_1:12;
A5: (S-min X)`2 = S-bound X by EUCLID:52;
  assume
A6: S-bound X < S-bound Y;
  then
A7: S-bound (X\/Y) = S-bound X by Th25;
  X c= X\/Y by XBOOLE_1:7;
  then S-min X in S-most(X\/Y) by A2,A7,A5,A4,SPRECT_2:11;
  then
A8: (S-min(X\/Y))`1 <= (S-min X)`1 by A2,PSCOMP_1:55;
  per cases by A3,XBOOLE_0:def 3;
  suppose
    S-min(X\/Y) in X;
    then S-min(X\/Y) in S-most X by A6,A1,Th25,SPRECT_2:11;
    then (S-min(X\/Y))`1 >= (S-min X)`1 by PSCOMP_1:55;
    then (S-min(X\/Y))`1 = (S-min X)`1 by A8,XXREAL_0:1;
    hence thesis by A6,A1,A5,Th25,TOPREAL3:6;
  end;
  suppose
    S-min(X\/Y) in Y;
    hence thesis by A6,A7,A1,PSCOMP_1:24;
  end;
end;
