reserve n for Nat;

theorem Th22:
  for C be compact non vertical non horizontal Subset of TOP-REAL
2 holds (N-min L~Cage(C,n))..Upper_Seq(C,n) < (N-max L~Cage(C,n))..Upper_Seq(C,
  n)
proof
  let C be compact non vertical non horizontal Subset of TOP-REAL 2;
  set Wmi = W-min L~Cage(C,n);
  set Nmi = N-min L~Cage(C,n);
  set Nma = N-max L~Cage(C,n);
  set Ema = E-max L~Cage(C,n);
  set Rot = Rotate(Cage(C,n),Wmi);
A1: L~Rot = L~Cage(C,n) by REVROT_1:33;
  then
A2: Ema in rng Rot by SPRECT_2:46;
  Wmi in rng Cage(C,n) by SPRECT_2:43;
  then Rot/.1 = Wmi by FINSEQ_6:92;
  then
A3: Nmi..Rot < Nma..Rot & Nma..Rot <= Ema..Rot by A1,SPRECT_5:24,25;
A4: Nma in rng Rot by A1,SPRECT_2:40;
  Nmi in rng Rot by A1,SPRECT_2:39;
  then Upper_Seq(C,n) = Rot-:Ema & Nmi..(Rot-:Ema) = Nmi..Rot by A2,A3,
JORDAN1E:def 1,SPRECT_5:3,XXREAL_0:2;
  hence thesis by A2,A3,A4,SPRECT_5:3;
end;
