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)))) + (N-bound (L~ (Cage (C,n)))) = (S-bound C) + (N-bound C)

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

thus (S-bound (L~ (Cage (C,n)))) + (N-bound (L~ (Cage (C,n)))) = (S-bound (L~ (Cage (C,n)))) + ((N-bound C) + (((N-bound C) - (S-bound C)) / (2 |^ n))) by JORDAN10:6

.= ((S-bound C) - (((N-bound C) - (S-bound C)) / (2 |^ n))) + ((N-bound C) + (((N-bound C) - (S-bound C)) / (2 |^ n))) by JORDAN1A:63

.= (S-bound C) + (N-bound C) ; :: thesis: verum

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

thus (S-bound (L~ (Cage (C,n)))) + (N-bound (L~ (Cage (C,n)))) = (S-bound (L~ (Cage (C,n)))) + ((N-bound C) + (((N-bound C) - (S-bound C)) / (2 |^ n))) by JORDAN10:6

.= ((S-bound C) - (((N-bound C) - (S-bound C)) / (2 |^ n))) + ((N-bound C) + (((N-bound C) - (S-bound C)) / (2 |^ n))) by JORDAN1A:63

.= (S-bound C) + (N-bound C) ; :: thesis: verum