reserve n for Nat;

theorem Th29:
  for C be compact connected non vertical non horizontal Subset of
  TOP-REAL 2 holds (S-min L~Cage(C,n))..Lower_Seq(C,n) <= (W-min 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 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: Wmi in rng Rot by SPRECT_2:43;
  Ema in rng Cage(C,n) by SPRECT_2:46;
  then Rot/.1 = Ema by FINSEQ_6:92;
  then
A4: Smi..Rot <= Wmi..Rot by A2,SPRECT_5:41;
  Smi in rng Rot by A2,SPRECT_2:41;
  then Smi..(Rot-:Wmi) = Smi..Rot by A3,A4,SPRECT_5:3;
  hence thesis by A1,A3,A4,SPRECT_5:3;
end;
