reserve n for Nat;

theorem Th41:
  for C be compact non vertical non horizontal Subset of TOP-REAL
  2 holds Rev Lower_Seq(C,n) is_a_h.c._for Cage(C,n)
proof
  let C be compact non vertical non horizontal Subset of TOP-REAL 2;
A1: (Rev Lower_Seq(C,n)/.1)`1 = (Lower_Seq(C,n)/.len Lower_Seq(C,n))`1 by
FINSEQ_5:65
    .= (W-min L~Cage(C,n))`1 by JORDAN1F:8
    .= W-bound L~Cage(C,n) by EUCLID:52;
A2: (Rev Lower_Seq(C,n)/.len Rev Lower_Seq(C,n))`1 = (Rev Lower_Seq(C,n)/.
  len Lower_Seq(C,n))`1 by FINSEQ_5:def 3
    .= (Lower_Seq(C,n)/.1)`1 by FINSEQ_5:65
    .= (E-max L~Cage(C,n))`1 by JORDAN1F:6
    .= E-bound L~Cage(C,n) by EUCLID:52;
  Rev Lower_Seq(C,n) is_in_the_area_of Cage(C,n) by JORDAN1E:18,SPRECT_3:51;
  hence thesis by A1,A2,SPRECT_2:def 2;
end;
