reserve n for Nat;

theorem
  for X,Y be non empty compact Subset of TOP-REAL 2 st E-bound X <
  E-bound Y holds E-min (X\/Y) = E-min Y
proof
  let X,Y be non empty compact Subset of TOP-REAL 2;
A1: (E-min(X\/Y))`1 = E-bound (X\/Y) by EUCLID:52;
A2: X\/Y is compact by COMPTS_1:10;
  then
A3: E-min(X\/Y) in X\/Y by SPRECT_1:14;
A4: E-min Y in Y by SPRECT_1:14;
A5: (E-min Y)`1 = E-bound Y by EUCLID:52;
  assume
A6: E-bound X < E-bound Y;
  then
A7: E-bound (X\/Y) = E-bound Y by Th24;
  Y c= X\/Y by XBOOLE_1:7;
  then E-min Y in E-most(X\/Y) by A2,A7,A5,A4,SPRECT_2:13;
  then
A8: (E-min(X\/Y))`2 <= (E-min Y)`2 by A2,PSCOMP_1:47;
  per cases by A3,XBOOLE_0:def 3;
  suppose
    E-min(X\/Y) in Y;
    then E-min(X\/Y) in E-most Y by A6,A1,Th24,SPRECT_2:13;
    then (E-min(X\/Y))`2 >= (E-min Y)`2 by PSCOMP_1:47;
    then (E-min(X\/Y))`2 = (E-min Y)`2 by A8,XXREAL_0:1;
    hence thesis by A6,A1,A5,Th24,TOPREAL3:6;
  end;
  suppose
    E-min(X\/Y) in X;
    hence thesis by A6,A7,A1,PSCOMP_1:24;
  end;
end;
