let n, m be Element of NAT ; :: thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds (E-bound (L~ (Cage C,n))) + (W-bound (L~ (Cage C,n))) = (E-bound (L~ (Cage C,m))) + (W-bound (L~ (Cage C,m)))
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); :: thesis: (E-bound (L~ (Cage C,n))) + (W-bound (L~ (Cage C,n))) = (E-bound (L~ (Cage C,m))) + (W-bound (L~ (Cage C,m)))
thus (E-bound (L~ (Cage C,n))) + (W-bound (L~ (Cage C,n))) = ((E-bound C) + (((E-bound C) - (W-bound C)) / (2 |^ n))) + (W-bound (L~ (Cage C,n))) by Th85
.= ((E-bound C) + (((E-bound C) - (W-bound C)) / (2 |^ n))) + ((W-bound C) - (((E-bound C) - (W-bound C)) / (2 |^ n))) by Th83
.= ((E-bound C) + (((E-bound C) - (W-bound C)) / (2 |^ m))) + ((W-bound C) - (((E-bound C) - (W-bound C)) / (2 |^ m)))
.= ((E-bound C) + (((E-bound C) - (W-bound C)) / (2 |^ m))) + (W-bound (L~ (Cage C,m))) by Th83
.= (E-bound (L~ (Cage C,m))) + (W-bound (L~ (Cage C,m))) by Th85 ; :: thesis: verum