let m, n be 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 Th64
.= ((E-bound C) + (((E-bound C) - (W-bound C)) / (2 |^ n))) + ((W-bound C) - (((E-bound C) - (W-bound C)) / (2 |^ n))) by Th62
.= ((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 Th62
.= (E-bound (L~ (Cage (C,m)))) + (W-bound (L~ (Cage (C,m)))) by Th64 ; :: thesis: verum