reserve n for Nat;

theorem
  for C be compact non vertical non horizontal Subset of TOP-REAL 2
  holds S-min L~Cage(C,n) in rng Lower_Seq(C,n) & S-min 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-min 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_2:41;
A3: Upper_Seq(C,n)/.1 = W-min L~Cage(C,n) by JORDAN1F:5;
A4: 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
A5: E-max L~Cage(C,n) in rng f by FINSEQ_6:90,SPRECT_2:43;
A6: 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
A7: Upper_Seq(C,n)/.1 = f/.1 by A5,FINSEQ_5:44;
  then
A8: (E-max L~f)..f < (E-min L~f)..f by A4,JORDAN1F:5,SPRECT_5:26;
A9: (E-min L~f)..f <= (S-max L~f)..f by A7,A4,JORDAN1F:5,SPRECT_5:27;
  per cases;
  suppose
    S-min L~f <> W-min L~f;
    then (E-min L~f)..f < (S-min L~f)..f by A7,A3,A4,A9,SPRECT_5:28,XXREAL_0:2;
    then x in rng(f:-p) by A1,A2,A6,A4,A8,FINSEQ_6:62,XXREAL_0:2;
    hence
A10: 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 A10;
  end;
  suppose
A11: S-min L~f = W-min L~f;
    Lower_Seq(C,n)/.(len Lower_Seq(C,n)) = W-min L~Cage(C,n) by JORDAN1F:8;
    hence
A12: x in rng Lower_Seq(C,n) by A4,A11,FINSEQ_6:168;
    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 A12;
  end;
end;
