let n, m be Element of NAT ; :: thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds (N-bound (L~ (Cage C,n))) + (S-bound (L~ (Cage C,n))) = (N-bound (L~ (Cage C,m))) + (S-bound (L~ (Cage C,m)))
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); :: thesis: (N-bound (L~ (Cage C,n))) + (S-bound (L~ (Cage C,n))) = (N-bound (L~ (Cage C,m))) + (S-bound (L~ (Cage C,m)))
thus (N-bound (L~ (Cage C,n))) + (S-bound (L~ (Cage C,n))) = ((N-bound C) + (((N-bound C) - (S-bound C)) / (2 |^ n))) + (S-bound (L~ (Cage C,n))) by JORDAN10:6
.= ((N-bound C) + (((N-bound C) - (S-bound C)) / (2 |^ n))) + ((S-bound C) - (((N-bound C) - (S-bound C)) / (2 |^ n))) by Th84
.= ((N-bound C) + (((N-bound C) - (S-bound C)) / (2 |^ m))) + ((S-bound C) - (((N-bound C) - (S-bound C)) / (2 |^ m)))
.= ((N-bound C) + (((N-bound C) - (S-bound C)) / (2 |^ m))) + (S-bound (L~ (Cage C,m))) by Th84
.= (N-bound (L~ (Cage C,m))) + (S-bound (L~ (Cage C,m))) by JORDAN10:6 ; :: thesis: verum