reserve i,j,k,n for Nat;

theorem Th1:
  for C being compact connected non vertical non horizontal Subset
  of TOP-REAL 2 holds (W-min L~Cage(C,n))..Cage(C,n) > 1
proof
  let C be compact connected non vertical non horizontal Subset of TOP-REAL 2;
A1: Cage(C,n)/.1 = N-min L~Cage(C,n) by JORDAN9:32;
  then 1 < (N-max L~Cage(C,n))..Cage(C,n) by SPRECT_2:69;
  then 1 < (E-max L~Cage(C,n))..Cage(C,n) by A1,SPRECT_2:70,XXREAL_0:2;
  then 1 < (E-min L~Cage(C,n))..Cage(C,n) by A1,SPRECT_2:71,XXREAL_0:2;
  then 1 < (S-max L~Cage(C,n))..Cage(C,n) by A1,SPRECT_2:72,XXREAL_0:2;
  then 1 < (S-min L~Cage(C,n))..Cage(C,n) by A1,SPRECT_2:73,XXREAL_0:2;
  hence thesis by A1,SPRECT_2:74,XXREAL_0:2;
end;
