reserve n for Nat;

theorem Th5:
  for C be compact connected non vertical non horizontal Subset of
  TOP-REAL 2 holds Lower_Seq(C,n) is_sequence_on Gauge(C,n)
proof
  let C be compact connected non vertical non horizontal Subset of TOP-REAL 2;
  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;
  E-max L~Cage(C,n) in rng Cage(C,n) by SPRECT_2:46;
  then E-max L~Cage(C,n) in rng Rotate(Cage(C,n),W-min L~Cage(C,n)) by
FINSEQ_6:90,SPRECT_2:43;
  then Rotate(Cage(C,n),W-min L~Cage(C,n)):-E-max L~Cage(C,n) is_sequence_on
  Gauge(C,n) by A1,Th3;
  hence thesis by JORDAN1E:def 2;
end;
