reserve n for Nat;

theorem Th12:
  for C be compact non vertical non horizontal Subset of TOP-REAL
  2 holds S-max L~Cage(C,n) in rng Lower_Seq(C,n) & S-max L~Cage(C,n) in L~
  Lower_Seq(C,n)
proof
  let C be compact non vertical non horizontal Subset of TOP-REAL 2;
  set x = S-max L~Cage(C,n);
  set p = E-max L~Cage(C,n);
  set f = Rotate(Cage(C,n),W-min L~Cage(C,n));
A1: rng f = rng Cage(C,n) by FINSEQ_6:90,SPRECT_2:43;
A2: x in rng Cage(C,n) by SPRECT_2:42;
A3: L~Cage(C,n) = L~f by REVROT_1:33;
  E-max L~Cage(C,n) in rng Cage(C,n) by SPRECT_2:46;
  then
A4: E-max L~Cage(C,n) in rng f by FINSEQ_6:90,SPRECT_2:43;
A5: p in rng Cage(C,n) by SPRECT_2:46;
  Upper_Seq(C,n) = f-:E-max L~Cage(C,n) by JORDAN1E:def 1;
  then
A6: Upper_Seq(C,n)/.1 = f/.1 by A4,FINSEQ_5:44;
  then
A7: (E-min L~f)..f <= (S-max L~f)..f by A3,JORDAN1F:5,SPRECT_5:27;
  (E-max L~f)..f < (E-min L~f)..f by A6,A3,JORDAN1F:5,SPRECT_5:26;
  then x in rng(f:-p) by A1,A2,A5,A3,A7,FINSEQ_6:62,XXREAL_0:2;
  hence
A8: x in rng Lower_Seq(C,n) by JORDAN1E:def 2;
  len Lower_Seq(C,n) >= 2 by TOPREAL1:def 8;
  then rng Lower_Seq(C,n) c= L~Lower_Seq(C,n) by SPPOL_2:18;
  hence thesis by A8;
end;
