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

theorem Th1:
  [i,j] in Indices Gauge(C,n) & [i+1,j] in Indices Gauge(C,n)
implies dist(Gauge(C,n)*(1,1),Gauge(C,n)*(2,1)) = Gauge(C,n)*(i+1,j)`1 - Gauge(
  C,n)*(i,j)`1
proof
  set G = Gauge(C,n);
  assume that
A1: [i,j] in Indices G and
A2: [i+1,j] in Indices G;
A3: j <= width G by A1,MATRIX_0:32;
  1 <= j by A1,MATRIX_0:32;
  then
A4: 1 <= width G by A3,XXREAL_0:2;
A5: len G >= 4 by JORDAN8:10;
  then 2 <= len G by XXREAL_0:2;
  then
A6: [2,1] in Indices G by A4,MATRIX_0:30;
A7: dist(G*(i,j),G*(i+1,j)) = (E-bound C - W-bound C)/2|^n by A1,A2,GOBRD14:10;
  1 <= len G by A5,XXREAL_0:2;
  then [1,1] in Indices G by A4,MATRIX_0:30;
  then dist(G*(1,1),G*(1+1,1)) = dist(G*(i,j),G*(i+1,j)) by A6,A7,GOBRD14:10
    .= G*(i+1,j)`1 - G*(i,j)`1 by A1,A2,GOBRD14:5;
  hence thesis;
end;
