reserve n for Nat;

theorem Th10:
  for C be compact non vertical non horizontal Subset of TOP-REAL
  2 for n be Nat holds len Upper_Seq(C,n) + len Lower_Seq(C,n) = len
  Cage(C,n)+1
proof
  let C be compact non vertical non horizontal Subset of TOP-REAL 2;
  let n be Nat;
  thus len Upper_Seq(C,n) + len Lower_Seq(C,n) = (E-max L~Cage(C,n))..Rotate(
  Cage(C,n),W-min L~Cage(C,n))+ len Lower_Seq(C,n) by Th8
    .= (E-max L~Cage(C,n))..Rotate(Cage(C,n),W-min L~Cage(C,n))+ (len Rotate
  (Cage(C,n),W-min L~Cage(C,n))- (E-max L~Cage(C,n))..Rotate(Cage(C,n),W-min L~
  Cage(C,n))+1) by Th9
    .= (E-max L~Cage(C,n))..Rotate(Cage(C,n),W-min L~Cage(C,n))+ len Rotate(
  Cage(C,n),W-min L~Cage(C,n))- (E-max L~Cage(C,n))..Rotate(Cage(C,n),W-min L~
  Cage(C,n))+1
    .= len Cage(C,n)+1 by FINSEQ_6:179;
end;
