reserve i, j, k, n for Nat,
  P for Subset of TOP-REAL 2,
  C for connected compact non vertical non horizontal Subset of TOP-REAL 2;

theorem Th6:
  N-bound L~Cage(C,n) = N-bound C + (N-bound C - S-bound C)/(2|^n)
proof
  set a = N-bound C, s = S-bound C, w = W-bound C, f = Cage(C,n), G = Gauge(C,
  n);
  consider i such that
A1: 1 <= i and
A2: i+1 <= len G and
A3: f/.1 = G*(i,width G) and
  f/.2 = G*(i+1,width G) and
  N-min C in cell(G,i,width G-'1) and
  N-min C <> G*(i,width G-'1) by JORDAN9:def 1;
A4: len G = width G by JORDAN8:def 1;
  4 <= len G by JORDAN8:10;
  then
A5: 1 <= len G by XXREAL_0:2;
  i+0 <= i+1 by XREAL_1:6;
  then i <= len G by A2,XXREAL_0:2;
  then
A6: [i,len G] in Indices G by A1,A4,A5,MATRIX_0:30;
A7: 2|^n <> 0 by NEWTON:83;
  thus N-bound L~f = (N-min L~f)`2 by EUCLID:52
    .= (f/.1)`2 by JORDAN9:32
    .= |[w+((E-bound C-w)/(2|^n))*(i-2), s+((a-s)/(2|^n))*(len G-2)]|`2 by A3
,A4,A6,JORDAN8:def 1
    .= s+((a-s)/(2|^n))*(len G-2) by EUCLID:52
    .= s+((a-s)/(2|^n))*(2|^n+3-2) by JORDAN8:def 1
    .= s+((a-s)/(2|^n))*2|^n+(a-s)/(2|^n)
    .= s+(a-s)+(a-s)/(2|^n) by A7,XCMPLX_1:87
    .= a+(a-s)/(2|^n);
end;
