reserve i,j,k,m,n for Nat,
  f for FinSequence of the carrier of TOP-REAL 2,
  G for Go-board;
reserve C for compact non vertical non horizontal non empty
  being_simple_closed_curve Subset of TOP-REAL 2,
  p for Point of TOP-REAL 2,
  i1, j1,i2,j2 for Nat;

theorem Th10:
  for C being connected compact non vertical non horizontal Subset
  of TOP-REAL 2 for n being Nat holds Upper_Seq(C,n) is_sequence_on
  Gauge(C,n)
proof
  let C be connected compact non vertical non horizontal Subset of TOP-REAL 2;
  let n be Nat;
  Cage(C,n) is_sequence_on Gauge(C,n) by JORDAN9:def 1;
  then
A1: Rotate(Cage(C,n),W-min L~Cage(C,n)) is_sequence_on Gauge(C,n) by
REVROT_1:34;
  Upper_Seq(C,n) = Rotate(Cage(C,n),W-min L~Cage(C,n))-:E-max L~Cage(C,n)
  by JORDAN1E:def 1
    .= 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))) by FINSEQ_5:def 1;
  hence thesis by A1,GOBOARD1:22;
end;
