reserve C for Simple_closed_curve,
  i, j, n for Nat,
  p for Point of TOP-REAL 2;

theorem Th2:
  [i,j] in Indices Gauge(C,n) & [i,j+1] in Indices Gauge(C,n)
implies dist(Gauge(C,n)*(1,1),Gauge(C,n)*(1,2)) = Gauge(C,n)*(i,j+1)`2 - Gauge(
  C,n)*(i,j)`2
proof
  set G = Gauge(C,n);
  assume that
A1: [i,j] in Indices G and
A2: [i,j+1] in Indices G;
A3: 1 <= j+1 by A2,MATRIX_0:32;
  len G >= 4 by JORDAN8:10;
  then
A4: 1 <= len G by XXREAL_0:2;
  2|^n + 3 >= 3 by NAT_1:11;
  then width G >= 3 by JORDAN1A:28;
  then 2 <= width G by XXREAL_0:2;
  then
A5: [1,2] in Indices G by A4,MATRIX_0:30;
  j + 1 <= width G by A2,MATRIX_0:32;
  then 1 <= width G by A3,XXREAL_0:2;
  then
A6: [1,1] in Indices G by A4,MATRIX_0:30;
  dist(G*(i,j),G*(i,j+1)) = (N-bound C - S-bound C)/2|^n by A1,A2,GOBRD14:9;
  then dist(G*(1,1),G*(1,1+1)) = dist(G*(i,j),G*(i,j+1)) by A6,A5,GOBRD14:9
    .= G*(i,j+1)`2 - G*(i,j)`2 by A1,A2,GOBRD14:6;
  hence thesis;
end;
