reserve n for Nat;

theorem Th21:
  for C be compact non vertical non horizontal Subset of TOP-REAL
2 holds (W-max L~Cage(C,n))..Upper_Seq(C,n) <= (N-min 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 Wma = W-max 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: Wma in rng Rot by SPRECT_2:44;
A3: Nmi in rng Rot by A1,SPRECT_2:39;
A4: Ema in rng Rot by A1,SPRECT_2:46;
  Wmi in rng Cage(C,n) by SPRECT_2:43;
  then
A5: Rot/.1 = Wmi by FINSEQ_6:92;
  then
A6: Wma..Rot <= Nmi..Rot by A1,SPRECT_5:23;
A7: Upper_Seq(C,n) = Rot-:Ema & Nma..Rot <= Ema..Rot by A1,A5,JORDAN1E:def 1
,SPRECT_5:25;
A8: Nmi..Rot < Nma..Rot by A1,A5,SPRECT_5:24;
  then Nmi..Rot < Ema..Rot by A1,A5,SPRECT_5:25,XXREAL_0:2;
  then Wma..(Rot-:Ema) = Wma..Rot by A2,A4,A6,SPRECT_5:3,XXREAL_0:2;
  hence thesis by A4,A6,A8,A7,A3,SPRECT_5:3,XXREAL_0:2;
end;
