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

theorem
  for p, q being Point of TOP-REAL 2, r being Real st dist(Gauge(
C,n)*(1,1),Gauge(C,n)*(1,2)) < r & dist(Gauge(C,n)*(1,1),Gauge(C,n)*(2,1)) < r
& p in cell (Gauge (C, n), i, j) & q in cell (Gauge (C, n), i, j) & 1 <= i & i+
1 <= len Gauge (C,n) & 1 <= j & j+1 <= width Gauge(C,n) holds dist (p,q) < 2 *
  r
proof
  set G = Gauge (C, n);
  let p, q be Point of TOP-REAL 2, r be Real;
  assume that
A1: dist(G*(1,1),G*(1,2)) < r and
A2: dist(G*(1,1),G*(2,1)) < r and
A3: p in cell (G, i, j) and
A4: q in cell (G, i, j) and
A5: 1 <= i and
A6: i+1 <= len G and
A7: 1 <= j and
A8: j+1 <= width G;
A9: p`1 <= G*(i+1,j)`1 by A3,A5,A6,A7,A8,JORDAN9:17;
A10: p`2 <= G*(i,j+1)`2 by A3,A5,A6,A7,A8,JORDAN9:17;
A11: G*(i,j)`2 <= p`2 by A3,A5,A6,A7,A8,JORDAN9:17;
  j <= j+1 by NAT_1:11;
  then
A12: j <= width G by A8,XXREAL_0:2;
  i <= i+1 by NAT_1:11;
  then
A13: i <= len G by A6,XXREAL_0:2;
  then
A14: [i,j] in Indices G by A5,A7,A12,MATRIX_0:30;
A15: q`2 <= G*(i,j+1)`2 by A4,A5,A6,A7,A8,JORDAN9:17;
A16: G*(i,j)`2 <= q`2 by A4,A5,A6,A7,A8,JORDAN9:17;
A17: q`1 <= G*(i+1,j)`1 by A4,A5,A6,A7,A8,JORDAN9:17;
A18: G*(i,j)`1 <= q`1 by A4,A5,A6,A7,A8,JORDAN9:17;
  1 <= j+1 by NAT_1:11;
  then [i,j+1] in Indices G by A5,A8,A13,MATRIX_0:30;
  then
A19: G*(i,j+1)`2 - G*(i,j)`2 < r by A1,A14,Th2;
  1 <= i+1 by NAT_1:11;
  then [i+1,j] in Indices G by A6,A7,A12,MATRIX_0:30;
  then G*(i+1,j)`1 - G*(i,j)`1 < r by A2,A14,Th1;
  then
A20: (G*(i+1,j)`1 - G*(i,j)`1 ) + ( G*(i,j+1)`2 - G*(i,j)`2 ) < r + r by A19,
XREAL_1:8;
  G*(i,j)`1 <= p`1 by A3,A5,A6,A7,A8,JORDAN9:17;
  then dist (p,q) <= (G*(i+1,j)`1 - G*(i,j)`1 ) + ( G*(i,j+1)`2 - G*(i,j)`2 )
  by A9,A11,A10,A18,A17,A16,A15,TOPREAL6:95;
  hence thesis by A20,XXREAL_0:2;
end;
