reserve n for Nat;

theorem
  for X,Y be non empty compact Subset of TOP-REAL 2 st N-bound X <
  N-bound Y holds N-max (X\/Y) = N-max Y
proof
  let X,Y be non empty compact Subset of TOP-REAL 2;
A1: (N-max(X\/Y))`2 = N-bound (X\/Y) by EUCLID:52;
A2: X\/Y is compact by COMPTS_1:10;
  then
A3: N-max(X\/Y) in X\/Y by SPRECT_1:11;
A4: N-max Y in Y by SPRECT_1:11;
A5: (N-max Y)`2 = N-bound Y by EUCLID:52;
  assume
A6: N-bound X < N-bound Y;
  then
A7: N-bound (X\/Y) = N-bound Y by Th23;
  Y c= X\/Y by XBOOLE_1:7;
  then N-max Y in N-most(X\/Y) by A2,A7,A5,A4,SPRECT_2:10;
  then
A8: (N-max(X\/Y))`1 >= (N-max Y)`1 by A2,PSCOMP_1:39;
  per cases by A3,XBOOLE_0:def 3;
  suppose
    N-max(X\/Y) in Y;
    then N-max(X\/Y) in N-most Y by A6,A1,Th23,SPRECT_2:10;
    then (N-max(X\/Y))`1 <= (N-max Y)`1 by PSCOMP_1:39;
    then (N-max(X\/Y))`1 = (N-max Y)`1 by A8,XXREAL_0:1;
    hence thesis by A6,A1,A5,Th23,TOPREAL3:6;
  end;
  suppose
    N-max(X\/Y) in X;
    hence thesis by A6,A7,A1,PSCOMP_1:24;
  end;
end;
