reserve n for Nat;

theorem Th27:
  for C be compact connected non vertical non horizontal Subset of
  TOP-REAL 2 holds (E-min L~Cage(C,n))..Lower_Seq(C,n) <= (S-max L~Cage(C,n))..
  Lower_Seq(C,n)
proof
  let C be compact connected non vertical non horizontal Subset of TOP-REAL 2;
  set Ema = E-max L~Cage(C,n);
  set Emi = E-min L~Cage(C,n);
  set Sma = S-max L~Cage(C,n);
  set Smi = S-min L~Cage(C,n);
  set Wmi = W-min L~Cage(C,n);
  set Rot = Rotate(Cage(C,n),Ema);
A1: Lower_Seq(C,n) = Rot-:Wmi by Th18;
A2: L~Rot = L~Cage(C,n) by REVROT_1:33;
  then
A3: Emi in rng Rot by SPRECT_2:45;
A4: Sma in rng Rot by A2,SPRECT_2:42;
A5: Wmi in rng Rot by A2,SPRECT_2:43;
  Ema in rng Cage(C,n) by SPRECT_2:46;
  then
A6: Rot/.1 = Ema by FINSEQ_6:92;
  then
A7: Emi..Rot <= Sma..Rot by A2,SPRECT_5:39;
A8: Smi..Rot <= Wmi..Rot by A2,A6,SPRECT_5:41;
A9: Sma..Rot < Smi..Rot by A2,A6,SPRECT_5:40;
  then Sma..Rot < Wmi..Rot by A2,A6,SPRECT_5:41,XXREAL_0:2;
  then Emi..(Rot-:Wmi) = Emi..Rot by A3,A5,A7,SPRECT_5:3,XXREAL_0:2;
  hence thesis by A1,A5,A7,A9,A8,A4,SPRECT_5:3,XXREAL_0:2;
end;
