let n be Nat; :: thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds S-bound (L~ (Cage (C,n))) < S-bound C

let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); :: thesis: S-bound (L~ (Cage (C,n))) < S-bound C

A1: 2 |^ n > 0 by NEWTON:83;

N-bound C > (S-bound C) + 0 by SPRECT_1:32;

then (N-bound C) - (S-bound C) > 0 by XREAL_1:20;

then A2: ((N-bound C) - (S-bound C)) / (2 |^ n) > (S-bound C) - (S-bound C) by A1, XREAL_1:139;

S-bound (L~ (Cage (C,n))) = (S-bound C) - (((N-bound C) - (S-bound C)) / (2 |^ n)) by JORDAN1A:63;

hence S-bound (L~ (Cage (C,n))) < S-bound C by A2, XREAL_1:11; :: thesis: verum

let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); :: thesis: S-bound (L~ (Cage (C,n))) < S-bound C

A1: 2 |^ n > 0 by NEWTON:83;

N-bound C > (S-bound C) + 0 by SPRECT_1:32;

then (N-bound C) - (S-bound C) > 0 by XREAL_1:20;

then A2: ((N-bound C) - (S-bound C)) / (2 |^ n) > (S-bound C) - (S-bound C) by A1, XREAL_1:139;

S-bound (L~ (Cage (C,n))) = (S-bound C) - (((N-bound C) - (S-bound C)) / (2 |^ n)) by JORDAN1A:63;

hence S-bound (L~ (Cage (C,n))) < S-bound C by A2, XREAL_1:11; :: thesis: verum