reserve n for Nat;

theorem Th33:
  for X,Y be non empty compact Subset of TOP-REAL 2 st W-bound X <
  W-bound Y holds W-min (X\/Y) = W-min X
proof
  let X,Y be non empty compact Subset of TOP-REAL 2;
A1: (W-min(X\/Y))`1 = W-bound (X\/Y) by EUCLID:52;
A2: X\/Y is compact by COMPTS_1:10;
  then
A3: W-min(X\/Y) in X\/Y by SPRECT_1:13;
A4: W-min X in X by SPRECT_1:13;
A5: (W-min X)`1 = W-bound X by EUCLID:52;
  assume
A6: W-bound X < W-bound Y;
  then
A7: W-bound (X\/Y) = W-bound X by Th26;
  X c= X\/Y by XBOOLE_1:7;
  then W-min X in W-most(X\/Y) by A2,A7,A5,A4,SPRECT_2:12;
  then
A8: (W-min(X\/Y))`2 <= (W-min X)`2 by A2,PSCOMP_1:31;
  per cases by A3,XBOOLE_0:def 3;
  suppose
    W-min(X\/Y) in X;
    then W-min(X\/Y) in W-most X by A6,A1,Th26,SPRECT_2:12;
    then (W-min(X\/Y))`2 >= (W-min X)`2 by PSCOMP_1:31;
    then (W-min(X\/Y))`2 = (W-min X)`2 by A8,XXREAL_0:1;
    hence thesis by A6,A1,A5,Th26,TOPREAL3:6;
  end;
  suppose
    W-min(X\/Y) in Y;
    hence thesis by A6,A7,A1,PSCOMP_1:24;
  end;
end;
