reserve i,j,k,m,n for Nat,
  f for FinSequence of the carrier of TOP-REAL 2,
  G for Go-board;

theorem Th8:
  for C be compact non vertical non horizontal Subset of TOP-REAL 2
  for n be Nat holds Lower_Seq(C,n)/.(len Lower_Seq(C,n)) = W-min L~
  Cage(C,n)
proof
  let C be compact non vertical non horizontal Subset of TOP-REAL 2;
  let n be Nat;
A1: W-min L~Cage(C,n) in rng Cage(C,n) by SPRECT_2:43;
  E-max L~Cage(C,n) in rng Cage(C,n) by SPRECT_2:46;
  then
  Lower_Seq(C,n) = Rotate(Cage(C,n),W-min L~Cage(C,n)):-E-max L~Cage(C,n)
  & E-max L~Cage(C,n) in rng Rotate(Cage(C,n),W-min L~Cage(C,n)) by FINSEQ_6:90
,JORDAN1E:def 2,SPRECT_2:43;
  hence
  Lower_Seq(C,n)/.(len Lower_Seq(C,n)) = Rotate(Cage(C,n),W-min L~Cage(C,
  n))/. (len Rotate(Cage(C,n),W-min L~Cage(C,n))) by FINSEQ_5:54
    .= Rotate(Cage(C,n),W-min L~Cage(C,n))/.1 by FINSEQ_6:def 1
    .= W-min L~Cage(C,n) by A1,FINSEQ_6:92;
end;
